Upper bounds for adjacent vertex-distinguishing edge coloring

2017 ◽  
Vol 35 (2) ◽  
pp. 454-462 ◽  
Author(s):  
Junlei Zhu ◽  
Yuehua Bu ◽  
Yun Dai
Keyword(s):  
Edge Coloring ◽  
Upper Bounds ◽  
Adjacent Vertex

Adjacent vertex distinguishing edge coloring of planar graphs without 3-cycles

2020 ◽  
Vol 12 (04) ◽  
pp. 2050035
Author(s):  
Danjun Huang ◽  
Xiaoxiu Zhang ◽  
Weifan Wang ◽  
Stephen Finbow
Keyword(s):  
Planar Graph ◽  
Planar Graphs ◽  
Edge Coloring ◽  
Discrete Math ◽  
Adjacent Vertex ◽  
Chromatic Index ◽  
Proper Edge Coloring ◽  
Minimum Number

The adjacent vertex distinguishing edge coloring of a graph [Formula: see text] is a proper edge coloring of [Formula: see text] such that the color sets of any pair of adjacent vertices are distinct. The minimum number of colors required for an adjacent vertex distinguishing edge coloring of [Formula: see text] is denoted by [Formula: see text]. It is observed that [Formula: see text] when [Formula: see text] contains two adjacent vertices of degree [Formula: see text]. In this paper, we prove that if [Formula: see text] is a planar graph without 3-cycles, then [Formula: see text]. Furthermore, we characterize the adjacent vertex distinguishing chromatic index for planar graphs of [Formula: see text] and without 3-cycles. This improves a result from [D. Huang, Z. Miao and W. Wang, Adjacent vertex distinguishing indices of planar graphs without 3-cycles, Discrete Math. 338 (2015) 139–148] that established [Formula: see text] for planar graphs without 3-cycles.

On the Adjacent Vertex Distinguishing Proper Edge Colorings of Several Classes of Complete 5-Partite Graphs

2013 ◽  
Vol 333-335 ◽  
pp. 1452-1455
Author(s):  
Chun Yan Ma ◽  
Xiang En Chen ◽  
Fang Yang ◽  
Bing Yao
Keyword(s):  
Chromatic Number ◽  
Edge Coloring ◽  
Adjacent Vertex ◽  
Chromatic Numbers ◽  
Edge Colorings ◽  
Proper Edge Coloring ◽  
Minimum Number

A proper $k$-edge coloring of a graph $G$ is an assignment of $k$ colors, $1,2,\cdots,k$, to edges of $G$. For a proper edge coloring $f$ of $G$ and any vertex $x$ of $G$, we use $S(x)$ denote the set of thecolors assigned to the edges incident to $x$. If for any two adjacent vertices $u$ and $v$ of $G$, we have $S(u)\neq S(v)$,then $f$ is called the adjacent vertex distinguishing proper edge coloring of $G$ (or AVDPEC of $G$ in brief). The minimum number of colors required in an AVDPEC of $G$ is called the adjacent vertex distinguishing proper edge chromatic number of $G$, denoted by $\chi^{'}_{\mathrm{a}}(G)$. In this paper, adjacent vertex distinguishing proper edge chromatic numbers of several classes of complete 5-partite graphs are obtained.

scholarly journals Adjacent vertex distinguishing edge coloring of the semistrong product of paths

2021 ◽  
Vol 1738 ◽  
pp. 012061
Author(s):  
AN Zhuo-mo ◽  
Jing Cai ◽  
Shuang-liang Tian
Keyword(s):  
Edge Coloring ◽  
Adjacent Vertex

scholarly journals Eternal Domination of Generalized Petersen Graph

2021 ◽  
Vol 2021 ◽  
pp. 1-10
Author(s):  
Ramy Shaheen ◽  
Ali Kassem
Keyword(s):  
Infinite Series ◽  
Dominating Set ◽  
Domination Number ◽  
Upper Bounds ◽  
Adjacent Vertex ◽  
Petersen Graph ◽  
Generalized Petersen Graph ◽  
Dominating Set Problem

An eternal dominating set of a graph G is a set of guards distributed on the vertices of a dominating set so that each vertex can be occupied by one guard only. These guards can defend any infinite series of attacks; an attack is defended by moving one guard along an edge from its position to the attacked vertex. We consider the “all guards move” of the eternal dominating set problem, in which one guard has to move to the attacked vertex and all the remaining guards are allowed to move to an adjacent vertex or stay in their current positions after each attack in order to form a dominating set on the graph and at each step can be moved after each attack. The “all guards move model” is called the m -eternal domination model. The size of the smallest m -eternal dominating set is called the m -eternal domination number and is denoted by γ m ∞ G . In this paper, we find γ m ∞ P n , 1 and γ m ∞ P n , 3 for n ≡ 0   mod   4 . We also find upper bounds for γ m ∞ P n , 2 and γ m ∞ P n , 3 when n is arbitrary.

scholarly journals Progress on the Adjacent Vertex Distinguishing Edge Coloring Conjecture

2020 ◽  
Vol 34 (4) ◽  
pp. 2221-2238
Author(s):  
Gwenaël Joret ◽  
William Lochet
Keyword(s):  
Edge Coloring ◽  
Adjacent Vertex

Adjacent vertex-distinguishing edge coloring of 2-degenerate graphs

2014 ◽  
Vol 31 (2) ◽  
pp. 874-880 ◽  
Author(s):  
Yi Wang ◽  
Jian Cheng ◽  
Rong Luo ◽  
Gregory Mulley
Keyword(s):  
Edge Coloring ◽  
Adjacent Vertex

scholarly journals THE PROPORTIONAL COLORING PROBLEM: OPTIMIZING BUFFERS IN RADIO MESH NETWORKS

2012 ◽  
Vol 04 (03) ◽  
pp. 1250028 ◽  
Author(s):  
FLORIAN HUC ◽  
CLÁUDIA LINHARES SALES ◽  
HERVÉ RIVANO
Keyword(s):  
Mesh Networks ◽  
Edge Coloring ◽  
Weighted Graph ◽  
Minimum Cost ◽  
Upper Bounds ◽  
Weighted Graphs ◽  
Chromatic Index ◽  
Lower And Upper Bounds ◽  
Coloring Problem ◽  
Wireless Mesh

In this paper, we consider a new edge coloring problem to model call scheduling optimization issues in wireless mesh networks: the proportional coloring. It consists in finding a minimum cost edge coloring of a graph which preserves the proportion given by the weights associated to each of its edges. We show that deciding if a weighted graph admits a proportional coloring is pseudo-polynomial while determining its proportional chromatic index is NP-hard. We then give lower and upper bounds for this parameter that can be computed in pseudo-polynomial time. We finally identify a class of graphs and a class of weighted graphs for which the proportional chromatic index can be exactly determined.

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