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Journal articles on the topic 'Rainbow subgraph'

Author: Grafiati

Published: 25 May 2024

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1

Axenovich, Maria, Tao Jiang, and Z. Tuza. "Local Anti-Ramsey Numbers of Graphs." Combinatorics, Probability and Computing 12, no. 5-6 (2003): 495–511. http://dx.doi.org/10.1017/s0963548303005868.

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A subgraph H in an edge-colouring is properly coloured if incident edges of H are assigned different colours, and H is rainbow if no two edges of H are assigned the same colour. We study properly coloured subgraphs and rainbow subgraphs forced in edge-colourings of complete graphs in which each vertex is incident to a large number of colours.

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2

Lestari, Dia, and I. Ketut Budayasa. "BILANGAN KETERHUBUNGAN PELANGI PADA PEWARNAAN-SISI GRAF." MATHunesa: Jurnal Ilmiah Matematika 8, no. 1 (2020): 25–34. http://dx.doi.org/10.26740/mathunesa.v8n1.p25-34.

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Let be a graph. An edge-coloring of is a function , where is a set of colors. Respect to a subgraph of is called a rainbow subgraph if all edges of get different colors. Graph is called rainbow connected if for every two distinct vertices of is joined by a rainbow path. The rainbow connection number of , denoted by , is the minimum number of colors needed in coloring all edges of such that is a rainbow connected. The main problem considered in this thesis is determining the rainbow connection number of graph. In this thesis, we determine the exact value of the rainbow connection number of some

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3

KOSTOCHKA, ALEXANDR, and MATTHEW YANCEY. "Large Rainbow Matchings in Edge-Coloured Graphs." Combinatorics, Probability and Computing 21, no. 1-2 (2012): 255–63. http://dx.doi.org/10.1017/s0963548311000605.

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Arainbow subgraphof an edge-coloured graph is a subgraph whose edges have distinct colours. Thecolour degreeof a vertexvis the number of different colours on edges incident withv. Wang and Li conjectured that fork≥ 4, every edge-coloured graph with minimum colour degreekcontains a rainbow matching of size at least ⌈k/2⌉. A properly edge-colouredK4has no such matching, which motivates the restrictionk≥ 4, but Li and Xu proved the conjecture for all other properly coloured complete graphs. LeSaulnier, Stocker, Wenger and West showed that a rainbow matching of size ⌊k/2⌋ is guaranteed to exist, a

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4

Hüffner, Falk, Christian Komusiewicz, Rolf Niedermeier, and Martin Rötzschke. "The Parameterized Complexity of the Rainbow Subgraph Problem." Algorithms 8, no. 1 (2015): 60–81. http://dx.doi.org/10.3390/a8010060.

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5

Matos Camacho, Stephan, Ingo Schiermeyer, and Zsolt Tuza. "Approximation algorithms for the minimum rainbow subgraph problem." Discrete Mathematics 310, no. 20 (2010): 2666–70. http://dx.doi.org/10.1016/j.disc.2010.03.032.

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6

Koch, Maria, Stephan Matos Camacho, and Ingo Schiermeyer. "Algorithmic approaches for the minimum rainbow subgraph problem." Electronic Notes in Discrete Mathematics 38 (December 2011): 765–70. http://dx.doi.org/10.1016/j.endm.2011.10.028.

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7

Gyárfás, András, Jenő Lehel, and Richard H. Schelp. "Finding a monochromatic subgraph or a rainbow path." Journal of Graph Theory 54, no. 1 (2006): 1–12. http://dx.doi.org/10.1002/jgt.20179.

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8

LOH, PO-SHEN, and BENNY SUDAKOV. "Constrained Ramsey Numbers." Combinatorics, Probability and Computing 18, no. 1-2 (2009): 247–58. http://dx.doi.org/10.1017/s0963548307008875.

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For two graphs S and T, the constrained Ramsey number f(S, T) is the minimum n such that every edge colouring of the complete graph on n vertices (with any number of colours) has a monochromatic subgraph isomorphic to S or a rainbow subgraph isomorphic to T. Here, a subgraph is said to be rainbow if all of its edges have different colours. It is an immediate consequence of the Erdős–Rado Canonical Ramsey Theorem that f(S, T) exists if and only if S is a star or T is acyclic. Much work has been done to determine the rate of growth of f(S, T) for various types of parameters. When S and T are bot

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9

Schiermeyer, Ingo. "On the minimum rainbow subgraph number of a graph." Ars Mathematica Contemporanea 6, no. 1 (2012): 83–88. http://dx.doi.org/10.26493/1855-3974.246.94d.

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10

Katrenič, Ján, and Ingo Schiermeyer. "Improved approximation bounds for the minimum rainbow subgraph problem." Information Processing Letters 111, no. 3 (2011): 110–14. http://dx.doi.org/10.1016/j.ipl.2010.11.005.

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11

Xiang, Changyuan, Yongxin Lan, Qinghua Yan, and Changqing Xu. "The Outer-Planar Anti-Ramsey Number of Matchings." Symmetry 14, no. 6 (2022): 1252. http://dx.doi.org/10.3390/sym14061252.

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A subgraph H of an edge-colored graph G is called rainbow if all of its edges have different colors. Let ar(G,H) denote the maximum positive integer t, such that there is a t-edge-colored graph G without any rainbow subgraph H. We denote by kK2 a matching of size k and On the class of all maximal outer-planar graphs on n vertices, respectively. The outer-planar anti-Ramsey number of graph H, denoted by ar(On,H), is defined as max{ar(On,H)|On∈On}. It seems nontrivial to determine the exact values for ar(On,H) because most maximal outer-planar graphs are asymmetry. In this paper, we obtain that

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12

Yuan, Chen, and Haibin Kan. "Revisiting a randomized algorithm for the minimum rainbow subgraph problem." Theoretical Computer Science 593 (August 2015): 154–59. http://dx.doi.org/10.1016/j.tcs.2015.05.042.

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13

Popa, Alexandru. "Better lower and upper bounds for the minimum rainbow subgraph problem." Theoretical Computer Science 543 (July 2014): 1–8. http://dx.doi.org/10.1016/j.tcs.2014.05.008.

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14

SUN, YUEFANG. "The (3, l)-Rainbow Edge-Index of Cartesian Product Graphs." Journal of Interconnection Networks 17, no. 03n04 (2017): 1741009. http://dx.doi.org/10.1142/s0219265917410092.

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For a graph G and a vertex subset [Formula: see text] of at least two vertices, an S-tree is a subgraph T of G that is a tree with [Formula: see text]. Two S-trees are said to be edge-disjoint if they have no common edge. Let [Formula: see text] denote the maximum number of edge-disjoint S-trees in G. For an integer K with [Formula: see text], the generalized k-edge-connectivity is defined as [Formula: see text]. An S-tree in an edge-colored graph is rainbow if no two edges of it are assigned the same color. Let [Formula: see text] and l be integers with [Formula: see text], the [Formula: see

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15

Mao, Yaping, and Yongtang Shi. "The complexity of determining the vertex-rainbow index of graphs." Discrete Mathematics, Algorithms and Applications 07, no. 04 (2015): 1550047. http://dx.doi.org/10.1142/s1793830915500470.

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The concept of the [Formula: see text]-rainbow index of a network comes from the communication of information between agencies of government, which was introduced by Chartrand et al. in 2010. As a natural counterpart of the [Formula: see text]-rainbow index, the concept of [Formula: see text]-vertex-rainbow index was also introduced. For a graph [Formula: see text] and a set [Formula: see text] of at least two vertices, an [Formula: see text]-Steiner tree or a Steiner tree connecting [Formula: see text] (or simply, an [Formula: see text]-tree) is such a subgraph [Formula: see text] of [Formula

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16

Tirodkar, Sumedh, and Sundar Vishwanathan. "On the Approximability of the Minimum Rainbow Subgraph Problem and Other Related Problems." Algorithmica 79, no. 3 (2017): 909–24. http://dx.doi.org/10.1007/s00453-017-0278-4.

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17

Goddard, Wayne, and Honghai Xu. "Vertex colorings without rainbow subgraphs." Discussiones Mathematicae Graph Theory 36, no. 4 (2016): 989. http://dx.doi.org/10.7151/dmgt.1896.

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18

Cui, Qing, Qinghai Liu, Colton Magnant, and Akira Saito. "Implications in rainbow forbidden subgraphs." Discrete Mathematics 344, no. 4 (2021): 112267. http://dx.doi.org/10.1016/j.disc.2020.112267.

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19

Holub, Přemysl, Zdeněk Ryjáček, Ingo Schiermeyer, and Petr Vrána. "Rainbow connection and forbidden subgraphs." Discrete Mathematics 338, no. 10 (2015): 1706–13. http://dx.doi.org/10.1016/j.disc.2014.08.008.

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20

Gorgol, Izolda. "Avoiding rainbow 2-connected subgraphs." Open Mathematics 15, no. 1 (2017): 393–97. http://dx.doi.org/10.1515/math-2017-0035.

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Abstract While defining the anti-Ramsey number Erdős, Simonovits and Sós mentioned that the extremal colorings may not be unique. In the paper we discuss the uniqueness of the colorings, generalize the idea of their construction and show how to use it to construct the colorings of the edges of complete split graphs avoiding rainbow 2-connected subgraphs. These colorings give the lower bounds for adequate anti-Ramsey numbers.

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21

Fujita, Shinya, and Colton Magnant. "Forbidden Rainbow Subgraphs That Force Large Highly Connected Monochromatic Subgraphs." SIAM Journal on Discrete Mathematics 27, no. 3 (2013): 1625–37. http://dx.doi.org/10.1137/120896906.

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22

Li, Wenjing, Xueliang Li, and Jingshu Zhang. "Rainbow vertex-connection and forbidden subgraphs." Discussiones Mathematicae Graph Theory 38, no. 1 (2018): 143. http://dx.doi.org/10.7151/dmgt.2004.

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23

Li, Wenjing, Xueliang Li, and Jingshu Zhang. "3-Rainbow Index and Forbidden Subgraphs." Graphs and Combinatorics 33, no. 4 (2017): 999–1008. http://dx.doi.org/10.1007/s00373-017-1783-6.

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24

Barrus, Michael D., Michael Ferrara, Jennifer Vandenbussche, and Paul S. Wenger. "Colored Saturation Parameters for Rainbow Subgraphs." Journal of Graph Theory 86, no. 4 (2017): 375–86. http://dx.doi.org/10.1002/jgt.22132.

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25

Kisielewicz, Andrzej, and Marek Szykuła. "Rainbow Induced Subgraphs in Proper Vertex Colorings." Fundamenta Informaticae 111, no. 4 (2011): 437–51. http://dx.doi.org/10.3233/fi-2011-572.

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26

Axenovich, Maria, and Perry Iverson. "Edge-colorings avoiding rainbow and monochromatic subgraphs." Discrete Mathematics 308, no. 20 (2008): 4710–23. http://dx.doi.org/10.1016/j.disc.2007.08.092.

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27

Wagner, Adam Zsolt. "Large Subgraphs in Rainbow-Triangle Free Colorings." Journal of Graph Theory 86, no. 2 (2017): 141–48. http://dx.doi.org/10.1002/jgt.22117.

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28

Rödl, Vojtech, and Zsolt Tuza. "Rainbow subgraphs in properly edge-colored graphs." Random Structures & Algorithms 3, no. 2 (1992): 175–82. http://dx.doi.org/10.1002/rsa.3240030207.

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29

Ma, Zhiqiang, Yaping Mao, Ingo Schiermeyer, and Meiqin Wei. "Complete bipartite graphs without small rainbow subgraphs." Discrete Applied Mathematics 346 (March 2024): 248–62. http://dx.doi.org/10.1016/j.dam.2023.12.020.

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30

Alon, Noga, Tao Jiang, Zevi Miller, and Dan Pritikin. "Properly colored subgraphs and rainbow subgraphs in edge-colorings with local constraints." Random Structures and Algorithms 23, no. 4 (2003): 409–33. http://dx.doi.org/10.1002/rsa.10102.

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31

Li, Xihe, and Ligong Wang. "Forbidden rainbow subgraphs that force large monochromatic or multicolored k-connected subgraphs." Discrete Applied Mathematics 285 (October 2020): 18–29. http://dx.doi.org/10.1016/j.dam.2020.05.004.

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32

Goddard, Wayne, and Robert Melville. "Coloring subgraphs with restricted amounts of hues." Open Mathematics 15, no. 1 (2017): 1171–80. http://dx.doi.org/10.1515/math-2017-0098.

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Abstract We consider vertex colorings where the number of colors given to specified subgraphs is restricted. In particular, given some fixed graph F and some fixed set A of positive integers, we consider (not necessarily proper) colorings of the vertices of a graph G such that, for every copy of F in G, the number of colors it receives is in A. This generalizes proper colorings, defective coloring, and no-rainbow coloring, inter alia. In this paper we focus on the case that A is a singleton set. In particular, we investigate the colorings where the graph F is a star or is 1-regular.

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33

Czap, Július. "Rainbow subgraphs in edge-colored planar and outerplanar graphs." Discrete Mathematics Letters 12 (June 23, 2023): 73–77. http://dx.doi.org/10.47443/dml.2023.048.

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34

Liu, Yuchen, and Yaojun Chen. "Rainbow subgraphs in Hamiltonian cycle decompositions of complete graphs." Discrete Mathematics 346, no. 8 (2023): 113479. http://dx.doi.org/10.1016/j.disc.2023.113479.

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35

Klavžar, Sandi, and Gašper Mekiš. "On the rainbow connection of Cartesian products and their subgraphs." Discussiones Mathematicae Graph Theory 32, no. 4 (2012): 783. http://dx.doi.org/10.7151/dmgt.1644.

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36

Brousek, Jan, Přemysl Holub, Zdeněk Ryjáček, and Petr Vrána. "Finite families of forbidden subgraphs for rainbow connection in graphs." Discrete Mathematics 339, no. 9 (2016): 2304–12. http://dx.doi.org/10.1016/j.disc.2016.02.015.

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37

Bradshaw, Peter. "Rainbow spanning trees in random subgraphs of dense regular graphs." Discrete Mathematics 347, no. 6 (2024): 113960. http://dx.doi.org/10.1016/j.disc.2024.113960.

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38

Ding, Jili, Hong Bian, and Haizheng Yu. "Anti-Ramsey Numbers in Complete k-Partite Graphs." Mathematical Problems in Engineering 2020 (September 7, 2020): 1–5. http://dx.doi.org/10.1155/2020/5136104.

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The anti-Ramsey number ARG,H is the maximum number of colors in an edge-coloring of G such that G contains no rainbow subgraphs isomorphic to H. In this paper, we discuss the anti-Ramsey numbers ARKp1,p2,…,pk,Tn, ARKp1,p2,…,pk,ℳ, and ARKp1,p2,…,pk,C of Kp1,p2,…,pk, where Tn,ℳ, and C denote the family of all spanning trees, the family of all perfect matchings, and the family of all Hamilton cycles in Kp1,p2,…,pk, respectively.

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39

Jahanbekam, Sogol, and Douglas B. West. "Rainbow Spanning Subgraphs of Small Diameter in Edge-Colored Complete Graphs." Graphs and Combinatorics 32, no. 2 (2015): 707–12. http://dx.doi.org/10.1007/s00373-015-1588-4.

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40

Alon, Noga, Alexey Pokrovskiy, and Benny Sudakov. "Random subgraphs of properly edge-coloured complete graphs and long rainbow cycles." Israel Journal of Mathematics 222, no. 1 (2017): 317–31. http://dx.doi.org/10.1007/s11856-017-1592-x.

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41

Holub, Přemysl, Zdeněk Ryjáček, and Ingo Schiermeyer. "On forbidden subgraphs and rainbow connection in graphs with minimum degree 2." Discrete Mathematics 338, no. 3 (2015): 1–8. http://dx.doi.org/10.1016/j.disc.2014.10.006.

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42

Cano, Pilar, Guillem Perarnau, and Oriol Serra. "Rainbow spanning subgraphs in bounded edge–colourings of graphs with large minimum degree." Electronic Notes in Discrete Mathematics 61 (August 2017): 199–205. http://dx.doi.org/10.1016/j.endm.2017.06.039.

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43

Jahanbekam, Sogol, and Douglas B. West. "Anti-Ramsey Problems for t Edge-Disjoint Rainbow Spanning Subgraphs: Cycles, Matchings, or Trees." Journal of Graph Theory 82, no. 1 (2015): 75–89. http://dx.doi.org/10.1002/jgt.21888.

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44

Jia, Yuxing, Mei Lu, and Yi Zhang. "Anti-Ramsey Problems in Complete Bipartite Graphs for t Edge-Disjoint Rainbow Spanning Subgraphs: Cycles and Matchings." Graphs and Combinatorics 35, no. 5 (2019): 1011–21. http://dx.doi.org/10.1007/s00373-019-02053-y.

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45

Yuster, Raphael. "Rainbow $H$-factors." Electronic Journal of Combinatorics 13, no. 1 (2006). http://dx.doi.org/10.37236/1039.

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An $H$-factor of a graph $G$ is a spanning subgraph of $G$ whose connected components are isomorphic to $H$. Given a properly edge-colored graph $G$, a rainbow $H$-subgraph of $G$ is an $H$-subgraph of $G$ whose edges have distinct colors. A rainbow $H$-factor is an $H$-factor whose components are rainbow $H$-subgraphs. The following result is proved. If $H$ is any fixed graph with $h$ vertices then every properly edge-colored graph with $hn$ vertices and minimum degree $(1-1/\chi(H))hn+o(n)$ has a rainbow $H$-factor.

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46

Ehard, Stefan, Stefan Glock, and Felix Joos. "A rainbow blow-up lemma for almost optimally bounded edge-colourings." Forum of Mathematics, Sigma 8 (2020). http://dx.doi.org/10.1017/fms.2020.38.

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Abstract A subgraph of an edge-coloured graph is called rainbow if all its edges have different colours. We prove a rainbow version of the blow-up lemma of Komlós, Sárközy, and Szemerédi that applies to almost optimally bounded colourings. A corollary of this is that there exists a rainbow copy of any bounded-degree spanning subgraph H in a quasirandom host graph G, assuming that the edge-colouring of G fulfills a boundedness condition that is asymptotically best possible. This has many applications beyond rainbow colourings: for example, to graph decompositions, orthogonal double covers, and

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47

LeSaulnier, Timothy D., Christopher Stocker, Paul S. Wenger, and Douglas B. West. "Rainbow Matching in Edge-Colored Graphs." Electronic Journal of Combinatorics 17, no. 1 (2010). http://dx.doi.org/10.37236/475.

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A rainbow subgraph of an edge-colored graph is a subgraph whose edges have distinct colors. The color degree of a vertex $v$ is the number of different colors on edges incident to $v$. Wang and Li conjectured that for $k\geq 4$, every edge-colored graph with minimum color degree at least $k$ contains a rainbow matching of size at least $\left\lceil k/2 \right\rceil$. We prove the slightly weaker statement that a rainbow matching of size at least $\left\lfloor k/2 \right\rfloor$ is guaranteed. We also give sufficient conditions for a rainbow matching of size at least $\left\lceil k/2 \right\rce

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48

He, Zhen, Peter Frankl, Ervin Győri, et al. "Extremal Results for Graphs Avoiding a Rainbow Subgraph." Electronic Journal of Combinatorics 31, no. 1 (2024). http://dx.doi.org/10.37236/11676.

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We say that $k$ graphs $G_1,G_2,\dots,G_k$ on a common vertex set of size $n$ contain a rainbow copy of a graph $H$ if their union contains a copy of $H$ with each edge belonging to a distinct $G_i$. We provide a counterexample to a conjecture of Frankl on the maximum product of the sizes of the edge sets of three graphs avoiding a rainbow triangle. We propose an alternative conjecture, which we prove under the additional assumption that the union of the three graphs is complete. Furthermore, we determine the maximum product of the sizes of the edge sets of three graphs or four graphs avoiding

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49

Axenovich, Maria, and Ryan Martin. "Avoiding Rainbow Induced Subgraphs in Vertex-Colorings." Electronic Journal of Combinatorics 15, no. 1 (2008). http://dx.doi.org/10.37236/736.

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For a fixed graph $H$ on $k$ vertices, and a graph $G$ on at least $k$ vertices, we write $G\longrightarrow H$ if in any vertex-coloring of $G$ with $k$ colors, there is an induced subgraph isomorphic to $H$ whose vertices have distinct colors. In other words, if $G\longrightarrow H$ then a totally multicolored induced copy of $H$ is unavoidable in any vertex-coloring of $G$ with $k$ colors. In this paper, we show that, with a few notable exceptions, for any graph $H$ on $k$ vertices and for any graph $G$ which is not isomorphic to $H$, $G\not\!\!\longrightarrow H$. We explicitly describe all

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50

Correia, David Munh, Alexey Pokrovskiy, and Benny Sudakov. "Short Proofs of Rainbow Matchings Results." International Mathematics Research Notices, July 17, 2022. http://dx.doi.org/10.1093/imrn/rnac180.

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Abstract A subgraph of an edge-coloured graph is called rainbow if all its edges have distinct colours. The study of rainbow subgraphs goes back to the work of Euler on Latin squares and has been the focus of extensive research ever since. Many conjectures in this area roughly say that “every edge coloured graph of a certain type contains a rainbow matching using every colour.” In this paper we introduce a versatile “sampling trick,” which allows us to asymptotically solve some well-known conjectures and to obtain short proofs of old results. In particular: $\bullet $ We give the first asympto

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