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Journal articles on the topic 'Prism of graphs'

Author: Grafiati

Published: 2 June 2025

Last updated: 31 July 2025

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1

Zeen El Deen, Mohamed R. "Enumeration of spanning trees in prisms of some graphs." Proyecciones (Antofagasta) 42, no. 2 (2023): 339–91. http://dx.doi.org/10.22199/issn.0717-6279-4664.

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In graph theory, a prism over a graph G is the cartesian product of the graph G with P₂. The purpose of this work is to investigate the complexity of the prisms of some path and cycle-related graphs. In particular, we obtain simpler and more explicit formulas for the complexity of a special class of prisms of path-related graphs: fan graph, ladder graph, the composition Pn[P₂] graph, and book graph. Moreover, we obtain straightforward formulas for the complexity of a special class of prisms of cycle-related graphs: wheel graph, gear graph, prism graph, n−crossed prism graph, mirror graph M(Cn)

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2

Raksha, M. R., P. Hithavarshini, Charles Dominic, and N. K. Sudev. "Injective coloring of complementary prism and generalized complementary prism graphs." Discrete Mathematics, Algorithms and Applications 12, no. 02 (2020): 2050026. http://dx.doi.org/10.1142/s1793830920500263.

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The complementary prism [Formula: see text] of a graph [Formula: see text] is the graph obtained by drawing edges between the corresponding vertices of a graph [Formula: see text] and its complement [Formula: see text]. In this paper, we generalize the concept of complementary prisms of graphs and determine the injective chromatic number of generalized complementary prisms of graphs. We prove that for any simple graph [Formula: see text] of order [Formula: see text], [Formula: see text] and if [Formula: see text] is a graph with a universal vertex, then [Formula: see text].

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3

Gayathri, M. "On Vertex-Based Dimension of Some Graphs Joining Certain Prism Graphs." Graduate Journal of Interdisciplinary Research, Reports and Reviews 2, no. 01 (2024): 46–53. https://doi.org/10.34256/gjir3.v2i01.12.

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Background: In graph theory, the prism graph is a type of graph that is characterised by having the structure of a prism as its underlying framework. The notion of a resolving set and that of metric dimension for a graph of a prism is important in uniquely identifying the vertices within a prism graph. For a non-trivial connected graph $\Gamma_{r}=\Gamma_{r}(V, E)$, an ordered subset $U$ of vertices $resolves$ any pair of different vertices $y_{1}, y_{2} \in V$, if $d(v, y_{1})\neq d(v, y_{2})$ for some $v\in U$. Such a set $U$ is said to be a resolving set for $\Gamma_{r}$ and the smallest ca

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4

Gomathi, P., and R. Murali. "LACEABILITY PROPERTIES IN PRISM GRAPHS." Advances and Applications in Discrete Mathematics 19, no. 4 (2018): 437–44. http://dx.doi.org/10.17654/dm019040437.

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5

Ortiz, Juan, Andrew Zemke, Hala King, Darren Narayan, and Mirko Horňák. "Minimalk-rankings for prism graphs." Involve, a Journal of Mathematics 3, no. 2 (2010): 183–90. http://dx.doi.org/10.2140/involve.2010.3.183.

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6

Aldred, R. E. L., and Michael D. Plummer. "Matching extension in prism graphs." Discrete Applied Mathematics 221 (April 2017): 25–32. http://dx.doi.org/10.1016/j.dam.2016.12.017.

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7

Weigang Sun, Shuai Wang, and Jingyuan Zhang. "COUNTING SPANNING TREES IN PRISM AND ANTI-PRISM GRAPHS." Journal of Applied Analysis & Computation 6, no. 1 (2016): 65–75. http://dx.doi.org/10.11948/2016006.

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8

Martinez, Paul, Juan Ortiz, Maggy Tomova, and Cindy Wyels. "Radio numbers for generalized prism graphs." Discussiones Mathematicae Graph Theory 31, no. 1 (2011): 45. http://dx.doi.org/10.7151/dmgt.1529.

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9

Jacoby, Liza, Ralph Morrison, and Ben Weber. "Prism graphs in tropical plane curves." Involve, a Journal of Mathematics 14, no. 3 (2021): 495–510. http://dx.doi.org/10.2140/involve.2021.14.495.

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10

He, Xiaocong, Yongtao Li, and Lihua Feng. "Extremal graphs for the odd prism." Discrete Mathematics 348, no. 1 (2025): 114249. http://dx.doi.org/10.1016/j.disc.2024.114249.

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11

Čangalović, Mirjana, Jozef Kratica, Vera Kovačević-Vujčić, and Milica Stojanović. "Minimal doubly resolving sets of prism graphs." Optimization 62, no. 8 (2013): 1037–43. http://dx.doi.org/10.1080/02331934.2013.772999.

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12

Uma, G., S. Amutha, N. Anbazhagan, and B. Koushick. "Exploring Prism Graphs with Fractional Domination Parameters." Mathematics and Statistics 12, no. 5 (2024): 448–54. http://dx.doi.org/10.13189/ms.2024.120506.

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13

Yingtaweesittikul, Hatairat, Sayan Panma, and Penying Rochanakul. "Formulas for the Number of Weak Homomorphisms from Paths to Ladder Graphs and Stacked Prism Graphs." Journal of Mathematics 2023 (December 11, 2023): 1–13. http://dx.doi.org/10.1155/2023/1159532.

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Let G and H be graphs. A mapping f from V G to V H is called a weak homomorphism from G to H if f x = f y or f x , f y ∈ E H whenever x , y ∈ E G . A ladder graph is the Cartesian product of two paths, where one of the paths has only one edge. A stacked prism graph is the Cartesian product of a path and a cycle. In this paper, we provide a formula to determine the number of weak homomorphisms from paths to ladder graphs and a formula to determine the number of weak homomorphisms from paths to stacked prism graphs.

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14

R. Sabitha. "Evaluating Star Vertex Cochromatic Number in Prism, Sunlet and Derived Graphs." Communications on Applied Nonlinear Analysis 31, no. 6s (2024): 252–58. http://dx.doi.org/10.52783/cana.v31.1219.

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In this discussion, the star cochromatic number q is found for the following graphs: Prism Graph q[Y_m], Line Graph of Prism Graph q[〖L(Y〗_m)], Middle Graph of Prism Graph q[〖M(Y〗_m)], Sunlet Graph q[S_m], Line Graph of Sunlet Graph q[L(S_m)], Middle Graph of Sunlet Graph q[M(S_m)].

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15

Rosicka, Monika. "Convex and weakly convex domination in prism graphs." Discussiones Mathematicae Graph Theory 39, no. 3 (2019): 741. http://dx.doi.org/10.7151/dmgt.2207.

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16

Irfan, Muhammad, Martin Baca, and Andrea Semanicova-Fenovcikova. "On reflexive edge strength of generalized prism graphs." Electronic Journal of Graph Theory and Applications 10, no. 2 (2022): 415. http://dx.doi.org/10.5614/ejgta.2022.10.2.6.

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17

Ayhan Ahmed Al-Shumam. "On the Domination Numbers of Certain Prism Graphs." Tikrit Journal of Pure Science 27, no. 1 (2022): 90–98. http://dx.doi.org/10.25130/tjps.v27i1.85.

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A dominating set S of a graph , is a subset of the vertex set V (G) such that any vertex not in S is adjacent to at least one vertex in S .The domination number of a graph G denoted by is the minimum size of the dominating sets of G. In this paper we introduced the domination numbers of certain prism graphs.

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18

Ellingham, M. N., Pouria Salehi Nowbandegani, and Songling Shan. "Toughness and prism-hamiltonicity of P4-free graphs." Discrete Applied Mathematics 284 (September 2020): 201–6. http://dx.doi.org/10.1016/j.dam.2020.03.035.

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19

Diot, Emilie, Marko Radovanović, Nicolas Trotignon, and Kristina Vušković. "The (theta, wheel)-free graphs Part I: Only-prism and only-pyramid graphs." Journal of Combinatorial Theory, Series B 143 (July 2020): 123–47. http://dx.doi.org/10.1016/j.jctb.2017.12.004.

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20

Arakawa, Riku, Hiromu Yakura, Vimal Mollyn, et al. "PrISM-Tracker." Proceedings of the ACM on Interactive, Mobile, Wearable and Ubiquitous Technologies 6, no. 4 (2022): 1–27. http://dx.doi.org/10.1145/3569504.

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A user often needs training and guidance while performing several daily life procedures, e.g., cooking, setting up a new appliance, or doing a COVID test. Watch-based human activity recognition (HAR) can track users' actions during these procedures. However, out of the box, state-of-the-art HAR struggles from noisy data and less-expressive actions that are often part of daily life tasks. This paper proposes PrISM-Tracker, a procedure-tracking framework that augments existing HAR models with (1) graph-based procedure representation and (2) a user-interaction module to handle model uncertainty.

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21

Paul, R. Joseph, and U. Mary. "Geo chromatic number of certain graphs." Journal of Interdisciplinary Mathematics 26, no. 1 (2023): 11–16. http://dx.doi.org/10.47974/jim-1643.

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22

Muaengwaeng, Artchariya, Ratinan Boonklurb, and Sirirat Singhun. "Pancyclicity of the n-Generalized Prism over Skirted Graphs." Symmetry 14, no. 4 (2022): 816. http://dx.doi.org/10.3390/sym14040816.

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A side skirt is a planar rooted tree T, T≠P2, where the root of T is a vertex of degree at least two, and all other vertices except the leaves are of degree at least three. A reduced Halin graph or a skirted graph is a plane graph G=T∪P, where T is a side skirt, and P is a path connecting the leaves of T in the order determined by the embedding of T. The structure of reduced Halin or skirted graphs contains both symmetry and asymmetry. For n≥2 and Pn=v1v2v3⋯vn as a path of length n−1, we call the Cartesian product of a graph G and a path Pn, the n-generalized prism over a graph G. We have know

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23

El-Kholy, E. M., and H. Ahmed. "Folding List of Graphs Obtained from a Given Graph." International Journal of Mathematics and Mathematical Sciences 2020 (November 23, 2020): 1–9. http://dx.doi.org/10.1155/2020/1316497.

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In this paper, we examine the relation between graph folding of a given graph and foldings of new graphs obtained from this graph by some techniques like dual, gear, subdivision, web, crown, simplex, crossed prism, and clique-sum graphs. In each case, we obtained the necessary and sufficient conditions, if exist, for these new graphs to be folded.

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24

Lihawa, Indrawati, Sumarno Ismail, Isran K. Hasan, Lailany Yahya, Salmun K. Nasib, and Nisky Imansyah Yahya. "Bilangan Terhubung Titik Pelangi pada Graf Hasil Operasi Korona Graf Prisma (P_(m,2)) dan Graf Lintasan (P_3)." Jambura Journal of Mathematics 4, no. 1 (2022): 145–51. http://dx.doi.org/10.34312/jjom.v4i1.11826.

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Rainbow vertex-connection number is the minimum k-coloring on the vertex graph G and is denoted by rvc(G). Besides, the rainbow-vertex connection number can be applied to some special graphs, such as prism graph and path graph. Graph operation is a method used to create a new graph by combining two graphs. Therefore, this research uses corona product operation to form rainbow-vertex connection number at the graph resulting from corona product operation of prism graph and path graph (Pm,2 P3) (P3 Pm,2). The results of this study obtain that the theorem of rainbow vertex-connection number at the

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25

Shaheen, Ramy, Ziad Kanaya, and Khaled Alshehada. "Game Chromatic Number of Generalized Petersen Graphs and Jahangir Graphs." Journal of Applied Mathematics 2020 (April 14, 2020): 1–4. http://dx.doi.org/10.1155/2020/6475427.

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Let G = V , E be a graph, and two players Alice and Bob alternate turns coloring the vertices of the graph G a proper coloring where no two adjacent vertices are signed with the same color. Alice's goal is to color the set of vertices using the minimum number of colors, which is called game chromatic number and is denoted by χ g G , while Bob's goal is to prevent Alice's goal. In this paper, we investigate the game chromatic number χ g G of Generalized Petersen Graphs G P n , k for k ≥ 3 and arbitrary n , n -Crossed Prism Graph, and Jahangir Graph J n , m .

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26

Redmon, Eric, Miles Mena, Megan Vesta, et al. "Optimal Tilings of Bipartite Graphs Using Self-Assembling DNA." PUMP Journal of Undergraduate Research 6 (March 13, 2023): 124–50. http://dx.doi.org/10.46787/pump.v6i0.2427.

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Motivated by the recent advancements in nanotechnology and the discovery of new laboratory techniques using the Watson-Crick complementary properties of DNA strands, formal graph theory has recently become useful in the study of self-assembling DNA complexes. Construction methods based on graph theory have resulted in significantly increased efficiency. We present the results of applying graph theoretical and linear algebra techniques for constructing crossed-prism graphs, crown graphs, book graphs, stacked book graphs, and helm graphs, along with kite, cricket, and moth graphs. In particular,

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27

Lisitsyna, M. A. "Perfect 3-colorings of prism and Möbius ladder graphs." Journal of Applied and Industrial Mathematics 7, no. 2 (2013): 215–20. http://dx.doi.org/10.1134/s1990478913020105.

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28

Hartvigsen, David, and Russell Mardon. "The prism-free planar graphs and their cycles bases." Journal of Graph Theory 15, no. 4 (1991): 431–41. http://dx.doi.org/10.1002/jgt.3190150408.

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29

Raza, Ali, Mobeen Munir, Tasawar Abbas, Sayed M. Eldin, and Ilyas Khan. "Spectrum of prism graph and relation with network related quantities." AIMS Mathematics 8, no. 2 (2022): 2634–47. http://dx.doi.org/10.3934/math.2023137.

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<abstract><p>Spectra of network related graphs have numerous applications in computer sciences, electrical networks and complex networks to explore structural characterization like stability and strength of these different real-world networks. In present article, our consideration is to compute spectrum based results of generalized prism graph which is well-known planar and polyhedral graph family belongs to the generalized Petersen graphs. Then obtained results are applied to compute some network related quantities like global mean-first passage time, average path length, number o

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30

Alfeche, Farene Loida, Victor Barraza, and Sergio Canoy. "Closeness Centrality of Vertices in Graphs Under Some Operations." European Journal of Pure and Applied Mathematics 16, no. 3 (2023): 1406–20. http://dx.doi.org/10.29020/nybg.ejpam.v16i3.4848.

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In this paper, we revisit the concept of (normalized) closeness centrality of a vertex in a graph and investigate it in some graphs under some operations. Specifically, we derive formulas that compute the closeness centrality of vertices in the shadow graph, complementary prism, edge corona, and disjunction of graphs.

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31

Zhang, Xiujun, Muhammad Ibrahim, Syed Bokhary, and Muhammad Siddiqui. "Edge Irregular Reflexive Labeling for the Disjoint Union of Gear Graphs and Prism Graphs." Mathematics 6, no. 9 (2018): 142. http://dx.doi.org/10.3390/math6090142.

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In graph theory, a graph is given names—generally a whole number—to edges, vertices, or both in a chart. Formally, given a graph G = ( V , E ) , a vertex naming is a capacity from V to an arrangement of marks. A diagram with such a capacity characterized defined is known as a vertex-marked graph. Similarly, an edge naming is a mapping of an element of E to an arrangement of marks. In this case, the diagram is called an edge-marked graph. We consider an edge irregular reflexive k-labeling for the disjoint association of wheel-related diagrams and deduce the correct estimation of the reflexive e

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32

Nugroho, Eri, and Kiki Ariyanti Sugeng. "On Local-Strong Rainbow Connection Numbers On Generalized Prism Graphs And Generalized Antiprism Graphs." Pattimura International Journal of Mathematics (PIJMath) 1, no. 2 (2022): 43–58. http://dx.doi.org/10.30598/pijmathvol1iss2pp43-58.

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Rainbow geodesic is the shortest path that connects two different vertices in graph such that every edge of the path has different colors. The strong rainbow connection number of a graph G, denoted by src(G), is the smallest number of colors required to color the edges of G such that there is a rainbow geodesic for each pair of vertices. The d-local strong rainbow connection number, denoted by lrscd, is the smallest number of colors required to color the edges of G such that any pair of vertices with a maximum distance d is connected by a rainbow geodesic. This paper contains some results of l

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33

Nagarathnamma, K. G., Leena N. Shenoy, and Sowmya Krishna. "Modified Detour Index of Hamiltonian Connected (Laceable) Graphs." Indian Journal Of Science And Technology 17, no. 19 (2024): 1923–34. http://dx.doi.org/10.17485/ijst/v17i19.1033.

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Objectives: To explore the bounds for the modified detour index of certain Hamiltonian connected and laceable graphs. Methods: The Wiener index , detour index and the modified detour index are used. Findings: Here we introduce the modified detour index and its least upper bounds for Hamiltonian connected and laceable graphs, by formulating the constraints. Novelty: Based on the modified detour index, the bounds for some special graphs such as: Hamiltonian connected graphs of two families of convex polytopes ( and ) and Hamiltonian laceable graphs of spider graph ( ) and image graph of prism gr

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34

Veeraraghavan, Prakash. "The G-Convexity and the G-Centroids of Composite Graphs." Mathematics 8, no. 11 (2020): 1927. http://dx.doi.org/10.3390/math8111927.

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The graph centroids defined through a topological property of a graph called g-convexity found its application in various fields. They have classified under the “facility location” problem. However, the g-centroid location for an arbitrary graph is NP-hard. Thus, it is necessary to devise an approximation algorithm for general graphs and polynomial-time algorithms for some special classes of graphs. In this paper, we study the relationship between the g-centroids of composite graphs and their factors under various well-known graph operations such as graph Joins, Cartesian products, Prism, and

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35

Triyani, Triyani, Mashuri Mashuri, Bunga Tirai Anarkis, and Slamet Riyadi. "The spectrum on prism graph using circulant matrix." Bulletin of Applied Mathematics and Mathematics Education 2, no. 1 (2022): 1–10. http://dx.doi.org/10.12928/bamme.v2i1.5129.

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Spectral graph theory discusses about the algebraic properties of graphs based on the spectrum of a graph. This article investigated the spectrum of prism graph. The method used in this research is the circulant matrix. The results showed that prism graph P2,s is a regular graph of degree 3, for s odd and s ≥ 3, P2,s is a circulantt graph with regular spectrum.

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36

Neethu, P. K., S. V. Ullas Chandran, Manoj Changat, and Sandi Klavžar. "On the General Position Number of Complementary Prisms." Fundamenta Informaticae 178, no. 3 (2021): 267–81. http://dx.doi.org/10.3233/fi-2021-2006.

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The general position number gp(G) of a graph G is the cardinality of a largest set of vertices S such that no element of S lies on a geodesic between two other elements of S. The complementary prism G G ¯ of G is the graph formed from the disjoint union of G and its complement G ¯ by adding the edges of a perfect matching between them. It is proved that gp(G G ¯ ) ≤ n(G) + 1 if G is connected and gp(G G ¯ ) ≤ n(G) if G is disconnected. Graphs G for which gp(G G ¯ ) = n(G) + 1 holds, provided that both G and G ¯ are connected, are characterized. A sharp lower bound on gp(G G ¯ ) is proved. If G

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37

McKee, Terry A. "Unique chords of unique cycles in 3-connected planar graphs." Discrete Mathematics, Algorithms and Applications 12, no. 04 (2020): 2050056. http://dx.doi.org/10.1142/s1793830920500561.

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Edges that are the unique chords of at least one cycle have been studied by a variety of authors over the past dozen years. This paper begins the study of those graphs in which each edge is the unique chord of exactly one cycle. The [Formula: see text]-connected planar graphs that enjoy this restriction are characterized by two infinite sequences (the dipyramid and trapezohedron [Formula: see text]-polytopes) together with three special graphs (the [Formula: see text]-antiprism, and the [Formula: see text]- and [Formula: see text]-prism graphs).

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38

Nasir, Ruby, Zohaib Zahid, and Sohail Zafar. "Edge version of metric dimension for the families of grid graphs and generalized prism graphs." Discrete Mathematics, Algorithms and Applications 12, no. 03 (2020): 2050037. http://dx.doi.org/10.1142/s1793830920500378.

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The minimum edge version of metric basis is the smallest set [Formula: see text] of edges in a connected graph [Formula: see text] such that for every pair of edges [Formula: see text] [Formula: see text][Formula: see text] there exists an edge [Formula: see text] [Formula: see text][Formula: see text] for which [Formula: see text] [Formula: see text] [Formula: see text] holds. In this paper, the families of grid graphs and generalized prism graphs have been studied for edge version of metric dimension. Edge version of metric dimension is found to be constant for both families of graphs.

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39

Mou, Gao, and Dmitrii V. Pasechnik. "Edge-dominating cycles, k-walks and Hamilton prisms in 2K2-free graphs." Journal of Knot Theory and Its Ramifications 25, no. 12 (2016): 1642011. http://dx.doi.org/10.1142/s0218216516420116.

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We show that an edge-dominating cycle in a [Formula: see text]-free graph can be found in polynomial time; this implies that every [Formula: see text]-tough [Formula: see text]-free graph admits a [Formula: see text]-walk, and it can be found in polynomial time. For this class of graphs, this proves a long-standing conjecture due to Jackson and Wormald [[Formula: see text]-walks of graphs, Australas. J. Combin. 2 (1990) 135–146]. Furthermore, we prove that for any [Formula: see text] every [Formula: see text]-tough [Formula: see text]-free graph is prism-Hamiltonian and give an effective const

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40

Jagannathan, M., Vernold Vivin.J, and Veninstine Vivik.J. "The Equitable Chromatic Bounds on Splitting of Block Circulant Graphs." Mathematical Problems in Engineering 2022 (November 8, 2022): 1–17. http://dx.doi.org/10.1155/2022/7603023.

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An equitable vertex coloring for the splitting of block circulant graphs is investigated. The block circulant graphs comprises block circulant matrices, where each block is itself a matrix. These blocks in each row are cyclically shifted one place to the right from those of the previous row. We approached such block circulant graphs in matrix representation and derived their independent sets using the neighbourhoods of each vertex. This classification makes the vertex coloring process to be simpler and equitable in most cases. In this framework, the equitable chromatic numbers are obtained for

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41

Nugroho, E., and K. A. Sugeng. "On d-local strong rainbow connection number of prism graphs." Journal of Physics: Conference Series 1722 (January 2021): 012053. http://dx.doi.org/10.1088/1742-6596/1722/1/012053.

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42

Nugroho, E., and K. A. Sugeng. "On d-local strong rainbow connection number of prism graphs." Journal of Physics: Conference Series 1722 (January 2021): 012053. http://dx.doi.org/10.1088/1742-6596/1722/1/012053.

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43

Mehmood, Tariq, H. Mahmood, and M. Hussain. "Edge Irregularity Strength of Multi Middle and Extended Prism Graphs." Journal of Computational and Theoretical Nanoscience 14, no. 11 (2017): 5248–52. http://dx.doi.org/10.1166/jctn.2017.6924.

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44

Cappelle, Márcia R., Erika M. M. Coelho, Les R. Foulds, and Humberto J. Longo. "Open-independent, Open-locating-dominating Sets in Complementary Prism Graphs." Electronic Notes in Theoretical Computer Science 346 (August 2019): 253–64. http://dx.doi.org/10.1016/j.entcs.2019.08.023.

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45

Ozeki, Kenta. "A degree sum condition for graphs to be prism hamiltonian." Discrete Mathematics 309, no. 13 (2009): 4266–69. http://dx.doi.org/10.1016/j.disc.2008.12.028.

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46

Chudnovsky, Maria, Frédéric Maffray, Paul Seymour, and Sophie Spirkl. "Even pairs and prism corners in square-free Berge graphs." Journal of Combinatorial Theory, Series B 131 (July 2018): 12–39. http://dx.doi.org/10.1016/j.jctb.2018.01.003.

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47

Špacapan, Simon. "A counterexample to prism-hamiltonicity of 3-connected planar graphs." Journal of Combinatorial Theory, Series B 146 (January 2021): 364–71. http://dx.doi.org/10.1016/j.jctb.2020.09.012.

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48

Bueno, Letícia R., and Peter Horák. "On hamiltonian cycles in the prism over the odd graphs." Journal of Graph Theory 68, no. 3 (2010): 177–88. http://dx.doi.org/10.1002/jgt.20550.

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49

Esther S, Jebisha, and Veninstine Vivik J. "The Locating and Local Locating Domination of Prism Family Graphs." Mathematics and Statistics 12, no. 3 (2024): 292–302. http://dx.doi.org/10.13189/ms.2024.120309.

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50

Bowling, Andrew, and Bryan Freyberg. "Type \((a,b,c)\) face-magic labelings of prism graphs." Utilitas Mathematica 122 (March 30, 2025): 93–108. https://doi.org/10.61091/um122-07.

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Abstract:

Let \(G=(V,E,F)\) be a planar graph with vertex set \(V\), edge set \(E\), and set of faces \(F.\) For nonnegative integers \(a,b,\) and \(c\), a type \((a,b,c)\) face-magic labeling of \(G\) is an assignment of \(a\) labels to each vertex, \(b\) labels to each edge, and \(c\) labels to each face from the set of integer labels \(\{1,2,\dots a|V|+b|E|+c|F|\}\) such that each label is used exactly once, and for each \(s\)-sided face \(f \in F,\) the sum of the label of \(f\) with the labels of the vertices and edges incident with \(f\) is equal to some fixed constant \(\mu_s\) for every \(s.\) W

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