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Journal articles on the topic 'Disjoint union'

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Published: 5 June 2025
Last updated: 15 July 2025

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1

Kling, Gerhard. "On the uniqueness of the Laplacian spectra of coalescence of complete graphs." Open Journal of Discrete Applied Mathematics 6, no. 1 (2023): 25–39. https://doi.org/10.30538/psrp-odam2023.0082.

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Using coalescence and cones, this study defines three types of graphs formed by amalgamating vertices of disjoint unions of complete graphs. The three types include the cone over a disjoint union of two complete graphs (C1), the cone over a disjoint union of \(k\) complete graphs (C2), and the \(l\) cone over a disjoint union of two complete graphs (C3). Coalescence of complete graphs (C1, C3) and the \(l\) cone (C3) are determined by their Laplacian spectra, a novel finding. Their Laplacian spectra reveal the size of the vertex cutset. Applications include the analysis of corporate networks, where individuals form coalescence of complete graphs through joint membership of two or more company boards.
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2

Mimura, Masato, and Hiroki Sako. "Group Approximation in Cayley Topology and Coarse Geometry, Part II: Fibred Coarse Embeddings." Analysis and Geometry in Metric Spaces 7, no. 1 (2019): 62–108. http://dx.doi.org/10.1515/agms-2019-0005.

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Abstract The objective of this series is to study metric geometric properties of disjoint unions of Cayley graphs of amenable groups by group properties of the Cayley accumulation points in the space of marked groups. In this Part II, we prove that a disjoint union admits a fibred coarse embedding into a Hilbert space (as a disjoint union) if and only if the Cayley boundary of the sequence in the space of marked groups is uniformly a-T-menable. We furthermore extend this result to ones with other target spaces. By combining our main results with constructions of Osajda and Arzhantseva–Osajda, we construct two systems of markings of a certain sequence of finite groups with two opposite extreme behaviors of the resulting two disjoint unions: With respect to one marking, the space has property A. On the other hand, with respect to the other, the space does not admit fibred coarse embeddings into Banach spaces with non-trivial type (for instance, uniformly convex Banach spaces) or Hadamard manifolds; the Cayley limit group is, furthermore, non-exact.
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3

Rajasekaran, G., and L. Uma. "ALPHA LABELINGS OF DISJOINT UNION OF HAIRY CYCLES." Ural Mathematical Journal 10, no. 1 (2024): 123. http://dx.doi.org/10.15826/umj.2024.1.011.

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In this paper, we prove the following results: 1) the disjoint union of \(n\geq 2\) isomorphic copies of the graph which is obtained by adding a pendent edge to each vertices of the cycle of order 4 admits \(\alpha\)-valuation; 2) the disjoint union of two isomorphic copies of the graph which is obtained by adding \(n\geq 1\) pendent edge to each vertices of the cycle of order 4 is admits \(\alpha\)-valuation; 3) the disjoint union of two isomorphic copies of the graph obtained by adding a pendent edge to each vertex of the cycle of order \(4m\) admits \(\alpha\)-valuation; 4) the disjoint union of two non-isomorphic copies of the graph obtained by adding a pendent edge to each vertices of the cycle of order \(4m\) and \(4m-2\) admits \(\alpha\)-valuation; 5) the disjoint union of two isomorphic copies of the graph which is obtained by adding a pendant edge to each vertex of the cycle of order \(4m-1(4m+2)\) is admitted graceful (\(\alpha\)-valuation).
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4

Elosmani, Aissa Omar, Djillali Bouagada, and Mohammed Chadli. "Graph-theoretic approach for structural controllability of two-dimensional linear systems." IMA Journal of Mathematical Control and Information 36, no. 3 (2018): 763–77. http://dx.doi.org/10.1093/imamci/dny003.

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Abstract The aim of this work is to present some criteria of structural-controllability relative to a certain structural two-dimensional linear systems. The most important property of two-dimensional systems is that the information is spread in two independent directions. To study their structural-controllability, we considered a new structural digraph associated with the different structural two-dimensional linear systems. The construction of this structural digraph is based on the disjoint-union graph notion, which translates the independence of the two dynamics, and consider separately the horizontal and vertical part of the state space model representation of two-dimensional systems. We define the two-dimensional disjoint-union digraph of structural Givone–Roesser model, and determine the two-dimensional disjoint-union digraph of structural Fornasini–Marchesini model, finally we derive a criteria based on the two-dimensional disjoint-union digraph.
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5

Slamin, Dafik, and Wyse Winnona. "Total Vertex Irregularity Strength of the Disjoint Union of Sun Graphs." International Journal of Combinatorics 2012 (August 3, 2012): 1–9. http://dx.doi.org/10.1155/2012/284383.

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A vertex irregular total -labeling of a graph with vertex set and edge set is an assignment of positive integer labels to both vertices and edges so that the weights calculated at vertices are distinct. The total vertex irregularity strength of , denoted by is the minimum value of the largest label over all such irregular assignment. In this paper, we consider the total vertex irregularity strengths of disjoint union of isomorphic sun graphs, , disjoint union of consecutive nonisomorphic sun graphs, , and disjoint union of any two nonisomorphic sun graphs .
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6

Xi, Jie. "Harmonious Labelings of Cj∪Ck∪Cl." Advanced Materials Research 304 (July 2011): 345–49. http://dx.doi.org/10.4028/www.scientific.net/amr.304.345.

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7

Sudibyo, Nugroho Arif, Ardymulya Iswardani, and Yohana Putra Surya Rahmad Hidayat. "Total Vertex Irregularity Strength of Disjoint Union of Ladder Rung Graph and Disjoint Union of Domino Graph." Jurnal Matematika "MANTIK" 6, no. 1 (2020): 47–51. http://dx.doi.org/10.15642/mantik.2020.6.1.47-51.

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We investigate a graph labeling called the total vertex irregularity strength (tvs(G)). A tvs(G) is minimum for which graph has a vertex irregular total -labeling. In this paper, we determine the total vertex irregularity strength of disjoint union of ladder rung graph and disjoint union of domino graph.
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8

Fraczak, Wojciech, Loukas Georgiadis, Andrew Miller, and Robert E. Tarjan. "Finding dominators via disjoint set union." Journal of Discrete Algorithms 23 (November 2013): 2–20. http://dx.doi.org/10.1016/j.jda.2013.10.003.

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9

Lucas, Joan M. "Postorder Disjoint Set Union is Linear." SIAM Journal on Computing 19, no. 5 (1990): 868–82. http://dx.doi.org/10.1137/0219060.

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10

Dukes, Peter, and Lea Howard. "Small maximally disjoint union-free families." Discrete Mathematics 308, no. 18 (2008): 4272–75. http://dx.doi.org/10.1016/j.disc.2007.08.006.

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11

Wang, Tao, Ming Ju Liu, and De Ming Li. "Magic Labeling of Disjoint Union Graphs." Acta Mathematica Sinica, English Series 35, no. 11 (2019): 1817–26. http://dx.doi.org/10.1007/s10114-019-8500-8.

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12

Abrams, Gene, and Mark Tomforde. "The prime spectrum and primitive ideal space of a graph C*-algebra." International Journal of Mathematics 25, no. 07 (2014): 1450070. http://dx.doi.org/10.1142/s0129167x14500700.

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We describe primitive and prime ideals in the C*-algebra C*(E) of a graph E satisfying Condition (K), together with the topologies on each of these spaces. In particular, we find that primitive ideals correspond to the set of maximal tails disjoint union the set of finite-return vertices, and that prime ideals correspond to the set of clusters of maximal tails disjoint union the set of finite-return vertices.
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13

NELSON, SAM, and CHAU-YIM WONG. "ON THE ORBIT DECOMPOSITION OF FINITE QUANDLES." Journal of Knot Theory and Its Ramifications 15, no. 06 (2006): 761–72. http://dx.doi.org/10.1142/s0218216506004701.

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We study the structure of finite quandles in terms of subquandles. Every finite quandle Q decomposes in a natural way as a union of disjoint Q-complemented subquandles; this decomposition coincides with the usual orbit decomposition of Q. Conversely, the structure of a finite quandle with a given orbit decomposition is determined by its structure maps. We describe an algorithm for finding quandle structures on a disjoint union of quandles.
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14

Bollobás, Béla, and Istvan Simon. "Probabilistic Analysis of Disjoint Set Union Algorithms." SIAM Journal on Computing 22, no. 5 (1993): 1053–74. http://dx.doi.org/10.1137/0222064.

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15

Atay, Arife. "Disjoint union of fuzzy soft topological spaces." AIMS Mathematics 8, no. 5 (2023): 10547–57. http://dx.doi.org/10.3934/math.2023535.

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<abstract><p>In this work, sums of fuzzy soft topological spaces are defined with free union of a pairwise disjoint non-empty family of fuzzy soft topological spaces. Firstly, we give general information of fuzzy soft topological spaces. Then, we define free union of fuzzy soft topological spaces and disjoint union topology of fuzzy soft topological spaces. We call the free union of a pairwise disjoint non-empty family of fuzzy soft topological spaces the sum of fuzzy soft topological spaces. We show what are the interchangeable properties between fuzzy soft topological spaces and the sum of fuzzy soft topological spaces. For example, there are fuzzy soft interior, fuzzy soft closure, fuzzy soft limit points. Also, we provide some properties showing the relationships between fuzzy soft topological spaces and their sums. Some of these are fuzzy soft base, fuzzy soft sequences, fuzzy soft connected-disconnected, fuzzy soft compact spaces. Also, part of the research for this article is work on fuzzy soft convergence on fuzzy soft topological sum. With this paper, some results, theorems and definitions for fuzzy soft topological sum have been acquired with the help of results, theorems and definitions given in previous studies about fuzzy soft topological spaces.</p></abstract>
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16

Chen, Juin-Han, and Hui-Chin Tang. "Representations of a Comparison Measure between Two Fuzzy Sets." Symmetry 12, no. 12 (2020): 2008. http://dx.doi.org/10.3390/sym12122008.

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This paper analyzes the representation behaviors of a comparison measure between two compared fuzzy sets. Three types of restrictions on two fuzzy sets are considered in this paper: two disjoint union fuzzy sets, two disjoint fuzzy sets and two general fuzzy sets. Differences exist among the numbers of possible representations of a comparison measure for the three types of fuzzy sets restrictions. The value of comparison measure is constant for two disjoint union fuzzy sets. There are 49 candidate representations of a comparison measure for two disjoint fuzzy sets, of which 13 candidate representations with one or two terms are obtained. For each candidate representation, a variant of the general axiomatic definition for a comparison measure is presented. Choosing the right candidate representation for a given application, we can easily and efficiently calculate and compare a comparison measure.
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17

Dahane, Intissar, Lobna Dridi, and Sami Lazaar. "F-n-resolvable spaces and compactifications." Applied General Topology 20, no. 1 (2019): 97. http://dx.doi.org/10.4995/agt.2019.10036.

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<p>A topological space is said to be resolvable if it is a union of<br />two disjoint dense subsets. More generally it is called n-resolvable if it is a union of n pairwise disjoint dense subsets. In this paper, we characterize topological spaces such that their reflections (resp., compactifications) are n-resolvable (resp., exactly-n-resolvable, strongly-exactly-n-resolvable), for some particular cases of reflections and compactifications.</p>
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18

Barack, Zeveliano Zidane, and Kiki Ariyanti Sugeng. "Modular irregularity strength of disjoint union of cycle-related graph." ITM Web of Conferences 61 (2024): 01001. http://dx.doi.org/10.1051/itmconf/20246101001.

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Let G = (V,E) be a graph with a vertex set V and an edge set E of G, with order n. Modular irregular labeling of a graph G is an edge k-labeling φ:E → {1, 2,…,k} such that the modular weight of all vertices is all different. The modular weight is defined by wtφ(u) = Σv∈N(u) φ(uv) (mod n). The minimum number k such that a graph G has modular irregular labeling with the largest label k is called modular irregularity strength of G. In this research, we determine the modular irregularity strength for a disjoint union of cycle graph, $ (mC_{n})=\frac{mn}{2}+1 $ for n ≡ 0 (mod 4), a disjoint union of sun graph, ms(m(Cn ⊙ K1))2 = ∞ for n and m even and ms(m(Cn ⊙ K1)) = mn otherwise, and a disjoint union of middle graph of cycle graph, ms(mM(Cn)) = ∞ for n and m both odd numbers and $(mM({C_n})) = {{mn} \over 2} + 1$ otherwise.
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19

Brendle, Jörg, and Yurii Khomskii. "Mad Families Constructed from Perfect Almost Disjoint Families." Journal of Symbolic Logic 78, no. 4 (2013): 1164–80. http://dx.doi.org/10.2178/jsl.7804070.

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AbstractWe prove the consistency of together with the existence of a -definable mad family, answering a question posed by Friedman and Zdomskyy in [7, Question 16]. For the proof we construct a mad family in L which is an ℵ1-union of perfect a.d. sets, such that this union remains mad in the iterated Hechler extension. The construction also leads us to isolate a new cardinal invariant, the Borel almost-disjointness number, defined as the least number of Borel a.d. sets whose union is a mad family. Our proof yields the consistency of (and hence, ).
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20

ALON, N., and P. FRANKL. "Families in Which Disjoint Sets Have Large Union." Annals of the New York Academy of Sciences 555, no. 1 Combinatorial (1989): 9–16. http://dx.doi.org/10.1111/j.1749-6632.1989.tb22431.x.

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21

Kamranian, Azam, and Ghaffar Raeisi. "Ramsey Number of Disjoint Union of Good Hypergraphs." Iranian Journal of Science and Technology, Transactions A: Science 44, no. 6 (2020): 1649–52. http://dx.doi.org/10.1007/s40995-020-00872-w.

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22

Comfort, W. W., and Akio Kato. "Non-homeomorphic Disjoint Spaces whose Union is $\o^*$." Rocky Mountain Journal of Mathematics 23, no. 2 (1993): 533–45. http://dx.doi.org/10.1216/rmjm/1181072575.

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23

Hu, Yi, Muhammad Awais Umar, Mustafa Habib, Ce Shi, Ghulam Rasool, and Zijiang Zhu. "Antimagicness of $mC_n$ -Path and Its Disjoint Union." IEEE Access 7 (2019): 155299–303. http://dx.doi.org/10.1109/access.2019.2949043.

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24

Lourdusamy, A., F. Joy Beaula, and F. Patrick. "On Tades of Disjoint Union of Some Graphs." Ars Combinatoria 157 (December 31, 2023): 89–94. http://dx.doi.org/10.61091/ars157-09.

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Consider a total labeling ξ of a graph G . For every two different edges e and f of G , let w t ( e ) ≠ w t ( f ) where weight of e = x y is defined as w t ( e ) = | ξ ( e ) – ξ ( x ) – ξ ( y ) | . Then ξ is called edge irregular total absolute difference k -labeling of G . Let k be the minimum integer for which there is a graph G with edge irregular total absolute difference labeling. This k is called the total absolute difference edge irregularity strength of the graph G , denoted t a d e s ( G ) . We compute t a d e s of S C n , disjoint union of grid and zigzag graph.
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25

Bae, Yongju, and Byeorhi Kim. "On sufficient conditions for being finite quandles." Journal of Knot Theory and Its Ramifications 28, no. 13 (2019): 1940011. http://dx.doi.org/10.1142/s021821651940011x.

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In 2006, Nelson and Wong introduced that every finite quandle can be decomposed into a disjoint union of some quandles and for given [Formula: see text] quandles satisfying certain conditions, there is a quandle which is a disjoint union of the [Formula: see text] quandles. In 2008, Ehrman, Gurpinar, Thibault and Yetter also introduced similar results about the decomposition of quandles. In this paper, we observe an operation table [Formula: see text] which consists of four sub-operation tables, where diagonal tables are quandle operations. We study which conditions make [Formula: see text] a quandle operation table.
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26

Preston, G. B. "Monogenic inverse semigroups." Journal of the Australian Mathematical Society. Series A. Pure Mathematics and Statistics 40, no. 3 (1986): 321–42. http://dx.doi.org/10.1017/s1446788700027543.

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AbstractWe give a survey of some of the realisations that have been given of monogenic inverse semigroups and discuss their relation to one another. We then analyse the representations by bijections, combined under composition, of monogenic inverse semigroups, and classify these into isomorphism types. This provides a particularly easy way of classifying monogenic inverse semigroups into isomorphism types. Of interest is that we find two quite distinct representations by bijections of free monogenic inverse semigroups and show that all such representations must contain one of these two representations. We call a bijection of the form ai ↦ ai+1, i = 1,2,…, r − 1, a finite link of length r, and one of the form ai ↦ ai+1, i = 1,2…, a forward link. The inverse of a forward link we call a backward link. Two bijections u: A → B and r: C → D are said to be strongly disjoint if A ∩ C, A ∩ D, B ∩ C and B ∩ D are each empty. The two distinct representations of a free monogenic inverse semigroup, that we have just referred to, are first, such that its generator is the union of a counbtable set os finite links that are pairwise storongly disjoint part of any representation of a free monogenic inverse semigroup, the remaining part not affecting the isomorphism type. Each representation of a monogenic inverse semigroup that is not free contains a strongly disjoint part, determining it to within isomorphism, that is generated by either the strongly disjoint union of a finite link and a permutation or the strongly disjoint union of a finite and a forward link.
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27

Calcut, Jack S., Jules R. Metcalf-Burton, Taylor J. Richard, and Liam T. Solus. "Borromean rays and hyperplanes." Journal of Knot Theory and Its Ramifications 23, no. 04 (2014): 1450020. http://dx.doi.org/10.1142/s0218216514500205.

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Three disjoint rays in ℝ3 form Borromean rays provided their union is knotted, but the union of any two components is unknotted. We construct infinitely many Borromean rays, uncountably many of which are pairwise inequivalent. We obtain uncountably many Borromean hyperplanes.
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28

Marsidi, Marsidi, and Ika Hesti Agustin. "The local antimagic on disjoint union of some family graphs." Jurnal Matematika "MANTIK" 5, no. 2 (2019): 69–75. http://dx.doi.org/10.15642/mantik.2019.5.2.69-75.

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A graph in this paper is nontrivial, finite, connected, simple, and undirected. Graph consists of a vertex set and edge set. Let u,v be two elements in vertex set, and q is the cardinality of edge set in G, a bijective function from the edge set to the first q natural number is called a vertex local antimagic edge labelling if for any two adjacent vertices and , the weight of is not equal with the weight of , where the weight of (denoted by ) is the sum of labels of edges that are incident to . Furthermore, any vertex local antimagic edge labelling induces a proper vertex colouring on where is the colour on the vertex . The vertex local antimagic chromatic number is the minimum number of colours taken over all colourings induced by vertex local antimagic edge labelling of . In this paper, we discuss about the vertex local antimagic chromatic number on disjoint union of some family graphs, namely path, cycle, star, and friendship, and also determine the lower bound of vertex local antimagic chromatic number of disjoint union graphs. The chromatic numbers of disjoint union graph in this paper attend the lower bound.
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29

Sali, Attila, and Klaus-Dieter Schewe. "A Characterisation of Coincidence Ideals for Complex Values." JUCS - Journal of Universal Computer Science 15, no. (1) (2009): 304–54. https://doi.org/10.3217/jucs-015-01-0304.

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We investigate properties of coincidence ideals in subattribute lattices that occur in complex value datamodels, i.e. sets of subattributes, on which two complex values coincide. We let complex values be defined by constructors for records, sets, multisets, lists, disjoint union and optionality, i.e. the constructors cover the gist of all complex value data models. Such lattices carry the structure of a Brouwer algebra as long as the union-constructor is absent, and for this case sufficient and necessary conditions for coincidence ideals are already known. In this paper, we extend the characterisation of coincidence ideals to the most general case. The presence of the disjoint union constructor complicates all results and proofs significantly. The reason for this is that the union-constructor causes non-trivial restructuring rules to hold. The characterisation of coincidence ideal is of decisive importance for the axiomatisation of (weak) functional dependencies.
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30

Goodman-Strauss, C. "Piecewise-smooth surfaces as the union of geodesic disks." Studia Scientiarum Mathematicarum Hungarica 41, no. 4 (2004): 453–56. http://dx.doi.org/10.1556/sscmath.41.2004.4.8.

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31

Nakić, Anamari, and Leo Storme. "Tight sets in finite classical polar spaces." Advances in Geometry 17, no. 1 (2017): 109–29. http://dx.doi.org/10.1515/advgeom-2016-0034.

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Abstract We show that every i-tight set in the Hermitian variety H(2r + 1, q) is a union of pairwise disjoint (2r + 1)-dimensional Baer subgeometries $\text{PG}(2r+1,\,\sqrt{q})$ and generators of H(2r + 1, q), if q ≥ 81 is an odd square and i < (q2/3 − 1)/2. We also show that an i-tight set in the symplectic polar space W(2r + 1, q) is a union of pairwise disjoint generators of W(2r + 1, q), pairs of disjoint r-spaces {Δ, Δ⊥}, and (2r + 1)-dimensional Baer subgeometries. For W(2r + 1, q) with r even, pairs of disjoint r-spaces {Δ, Δ⊥} cannot occur. The (2r + 1)-dimensional Baer subgeometries in the i-tight set of W(2r + 1, q) are invariant under the symplectic polarity ⊥ of W(2r + 1, q) or they arise in pairs of disjoint Baer subgeometries corresponding to each other under ⊥. This improves previous results where $i \lt q^{5/8} / \sqrt{2} +1$ was assumed. Generalizing known techniques and using recent results on blocking sets and minihypers, we present an alternative proof of this result and consequently improve the upper bound on i to (q2/3 − 1)/2. We also apply our results on tight sets to improve a known result on maximal partial spreads in W(2r + 1, q).
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Apituley, Fredrylo Alberth Noel Joddy, Mozart W. Talakua, and Yopi Andry Lesnussa. "ON THE IRREGULARITY STRENGTH AND MODULAR IRREGULARITY STRENGTH OF FRIENDSHIP GRAPHS AND ITS DISJOINT UNION." BAREKENG: Jurnal Ilmu Matematika dan Terapan 16, no. 3 (2022): 869–76. http://dx.doi.org/10.30598/barekengvol16iss3pp869-876.

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For a simple, undirected graph G with, at most one isolated vertex and no isolated edges, a labeling f:E(G)→{1,2,…,k1} of positive integers to the edges of G is called irregular if the weights of each vertex of G has a different value. The integer k1 is then called the irregularity strength of G. If the number of vertices in G or the order of G is |G|, then the labeling μ:E(G)→{1,2,…,k2} is called modular irregular if the remainder of the weights of each vertex of G divided by |G| has a different value. The integer k2 is then called the modular irregularity strength of G. The disjoint union of two or more graphs, denoted by ‘+’, is an operation where the vertex and edge set of the result each be the disjoint union of the vertex and edge sets of the given graphs. This study discusses about the irregularity and modular irregularity strength of friendship graphs and some of its disjoint union, The result given is s(Fm ) = m + 1, ms(Fm ) = m + 1 and ms(rFm ) = rm + ⌈r/2⌉, where r denotes the number of copies of friendship graphs
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33

Du, Yudan, Xiaohong Zhang, and Xiaogang An. "Transposition Regular AG-Groupoids and Their Decomposition Theorems." Mathematics 10, no. 9 (2022): 1396. http://dx.doi.org/10.3390/math10091396.

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In this paper, we introduce transposition regularity into AG-groupoids, and a variety of transposition regular AG-groupoids (L1/R1/LR, L2/R2/L3/R3-groupoids) are obtained. Their properties and structures are discussed by their decomposition theorems: (1) L1/R1-transposition regular AG-groupoids are equivalent to each other, and they can be decomposed into the union of disjoint Abelian subgroups; (2) L1/R1-transposition regular AG-groupoids are LR-transposition regular AG-groupoids, and an example is given to illustrate that not every LR-transposition regular AG-groupoid is an L1/R1-transposition regular AG-groupoid; (3) an AG-groupoid is an L1/R1-transposition regular AG-groupoid if it is an LR-transposition regular AG-groupoid satisfying a certain condition; (4) strong L2/R3-transposition regular AG-groupoids are equivalent to each other, and they are union of disjoint Abelian subgroups; (5) strong L3/R2-transposition regular AG-groupoids are equivalent to each other and they can be decomposed into union of disjoint AG subgroups. Their relations are discussed. Finally, we introduce various transposition regular AG-groupoid semigroups and discuss the relationships among them and the commutative Clifford semigroup as well as the Abelian group.
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BALISTER, PAUL. "Packing Circuits into KN." Combinatorics, Probability and Computing 10, no. 6 (2001): 463–99. http://dx.doi.org/10.1017/s0963548301004771.

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We shall pack circuits of arbitrary lengths into the complete graph KN. More precisely, if N is odd and [sum ]ti=1mi = (N2), mi [ges ] 3, then the edges of KN can be written as an edge-disjoint union of circuits of lengths m1,…,mt. Since the degrees of the vertices in any such packing must be even, this result cannot hold for even N. For N even, we prove that if [sum ]ti=1mi [les ] (N2) − N−2 then we can write some subgraph of KN as an edge-disjoint union of circuits of lengths m1,…,mt. In particular, KN minus a 1-factor can be written as a union of such circuits when [sum ]ti=1mi = (N2) − N−2. We shall also show that these results are best possible.
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35

Mio, Washington. "On boundary-link cobordism." Mathematical Proceedings of the Cambridge Philosophical Society 101, no. 2 (1987): 259–66. http://dx.doi.org/10.1017/s0305004100066615.

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An n-dimensional m-component link is an oriented smooth submanifold Σn of Sn+2, where is the ordered disjoint union of m submanifolds of Sn+2, each homeomorphic to Sn. Σ is a boundary link if there is an oriented smooth submanifold Vn+1 of Sn+1, the disjoint union of the submanifolds , such that ∂Vi = Σi (i = 1,…, m). A pair (Σ, V), where Σ is a boundary link and V as above, with each Vi connected (i = 1,…, m), is called an n-dimensional special Seifert pair. In this paper, we define a notion of cobordism of special Seifert pairs and give an algebraic description of the set (group) of cobordism classes.
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AbuGhneim, Omar A., and Baha' Abughazaleh. "Prime Labeling of Union of Some Graphs." European Journal of Pure and Applied Mathematics 17, no. 4 (2024): 3557–66. http://dx.doi.org/10.29020/nybg.ejpam.v17i4.5336.

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A prime labeling of a graph G is a map from the vertex set of G to the set {1, 2, ..., |V (G)|} such that any two adjacent vertices in the graph G have labels that are relatively prime. In this paper, we discuss when the disjoint union of some graphs is a prime graph.
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37

Hou, Duo, and Bruce Torrence. "Involutions Fixing the Disjoint Union of Odd-Dimensional Projective Spaces." Canadian Mathematical Bulletin 37, no. 1 (1994): 66–74. http://dx.doi.org/10.4153/cmb-1994-010-9.

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38

Moran, R., N. Bora, A. K. Baruah, and A. Bharali. "ST-COLORING OF JOIN AND DISJOINT UNION OF GRAPHS." Advances in Mathematics: Scientific Journal 9, no. 11 (2020): 9393–99. http://dx.doi.org/10.37418/amsj.9.11.44.

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39

El-Shanawany, R., A. El-Mesady, and S. M. Shaaban. "Mutually orthogonal graph squares for disjoint union of paths." Applied Mathematical Sciences 12, no. 7 (2018): 303–10. http://dx.doi.org/10.12988/ams.2018.8112.

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40

Ahmad, Ali, Martin Baca, and Muhammad Numan. "On irregularity strength of disjoint union of friendship graphs." Electronic Journal of Graph Theory and Applications 1, no. 2 (2013): 100–108. http://dx.doi.org/10.5614/ejgta.2013.1.2.3.

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41

Haemers, Willem H., and Mohammad Reza Oboudi. "Universal spectra of the disjoint union of regular graphs." Linear Algebra and its Applications 606 (December 2020): 244–48. http://dx.doi.org/10.1016/j.laa.2020.07.033.

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42

Zhang, Zhijun, Muhammad Awais Umar, Xiaojun Ren, Basharat Rehman Ali, Mujtaba Hussain, and Xiangmei Li. "Tree-Antimagicness of Web Graphs and Their Disjoint Union." Mathematical Problems in Engineering 2020 (April 9, 2020): 1–6. http://dx.doi.org/10.1155/2020/4565829.

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In graph theory, the graph labeling is the assignment of labels (represented by integers) to edges and/or vertices of a graph. For a graph G=V,E, with vertex set V and edge set E, a function from V to a set of labels is called a vertex labeling of a graph, and the graph with such a function defined is called a vertex-labeled graph. Similarly, an edge labeling is a function of E to a set of labels, and in this case, the graph is called an edge-labeled graph. In this research article, we focused on studying super ad,d-T4,2-antimagic labeling of web graphs W2,n and isomorphic copies of their disjoint union.
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43

Galil, Zvi, and Giuseppe F. Italiano. "Data structures and algorithms for disjoint set union problems." ACM Computing Surveys 23, no. 3 (1991): 319–44. http://dx.doi.org/10.1145/116873.116878.

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44

Bielak, Halina. "Ramsey numbers for a disjoint union of good graphs." Discrete Mathematics 310, no. 9 (2010): 1501–5. http://dx.doi.org/10.1016/j.disc.2009.12.001.

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45

Bielak, Halina. "Ramsey numbers for a disjoint union of some graphs." Applied Mathematics Letters 22, no. 4 (2009): 475–77. http://dx.doi.org/10.1016/j.aml.2008.06.018.

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Gao, Hai, Weisi Lin, Ping Xue, and Wan-Chi Siu. "Marker-based image segmentation relying on disjoint set union." Signal Processing: Image Communication 21, no. 2 (2006): 100–112. http://dx.doi.org/10.1016/j.image.2005.06.008.

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47

Li, Ping. "Planar Turán number of the disjoint union of cycles." Discrete Applied Mathematics 342 (January 2024): 260–74. http://dx.doi.org/10.1016/j.dam.2023.09.021.

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48

Shen, Zijie, Ruixiang Li, and Junping Shi. "A Study of Disjoint Set Union in Programming Competitions." Journal of Big Data and Computing 1, no. 4 (2023): 1–5. http://dx.doi.org/10.62517/jbdc.202301401.

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Disjoint Set Union (DSU) is a tree data structure, which is used to effectively deal with the problems of merging and querying disjoint sets. The DSU algorithm can be used to merge sets and query to which set the node belongs. The DSU algorithm is usually implemented using arrays and tree structures, but there are also methods that use hash tables. The DSU algorithm has a wide range of applications in the connectivity of graph theory, social network and image processing. It helps researchers better understand and analyze set operations, provides a basis for subsequent work, and promotes the solution and optimization of various problems in the field. At the same time, in the programming competition of college students, the use of DSU is more frequent, usually the use of set query is more, resulting in high time complexity and unable to solve the problem quickly. Therefore, path compression strategy is introduced to significantly improve the efficiency of set query. Finally, the application of path compression in union search is introduced in the form of programming competition.
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49

Koch, Sebastian. "About Graph Sums." Formalized Mathematics 29, no. 4 (2021): 249–78. http://dx.doi.org/10.2478/forma-2021-0023.

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Wu, Xiaoying, and Xiaohong Zhang. "The Decomposition Theorems of AG-Neutrosophic Extended Triplet Loops and Strong AG-(l, l)-Loops." Mathematics 7, no. 3 (2019): 268. http://dx.doi.org/10.3390/math7030268.

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In this paper, some new properties of Abel Grassmann‘s Neutrosophic Extended Triplet Loop (AG-NET-Loop) were further studied. The following important results were proved: (1) an AG-NET-Loop is weakly commutative if, and only if, it is a commutative neutrosophic extended triplet (NETG); (2) every AG-NET-Loop is the disjoint union of its maximal subgroups. At the same time, the new notion of Abel Grassmann’s (l, l)-Loop (AG-(l, l)-Loop), which is the Abel-Grassmann’s groupoid with the local left identity and local left inverse, were introduced. The strong AG-(l, l)-Loops were systematically analyzed, and the following decomposition theorem was proved: every strong AG-(l, l)-Loop is the disjoint union of its maximal sub-AG-groups.
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