On the precise value of the strong chromatic index of a planar graph with a large girth

The strong chromatic index of a graph $G$, denoted $\chi'_s(G)$, is the least number of colors needed to edge-color $G$ so that edges at distance at most two receive distinct colors. The strong list chromatic index, denoted $\chi'_{s,\ell}(G)$, is the least integer $k$ such that if arbitrary lists of size $k$ are assigned to each edge then $G$ can be edge-colored from those lists where edges at distance at most two receive distinct colors.We use the discharging method, the Combinatorial Nullstellensatz, and computation to show that if $G$ is a subcubic planar graph with ${\rm girth}(G) \geq 41$ then $\chi'_{s,\ell}(G) \leq 5$, answering a question of Borodin and Ivanova [Precise upper bound for the strong edge chromatic number of sparse planar graphs, Discuss. Math. Graph Theory, 33(4), (2014) 759--770]. We further show that if $G$ is a subcubic planar graph and ${\rm girth}(G) \geq 30$, then $\chi_s'(G) \leq 5$, improving a bound from the same paper.Finally, if $G$ is a planar graph with maximum degree at most four and ${\rm girth}(G) \geq 28$, then $\chi'_s(G)N \leq 7$, improving a more general bound of Wang and Zhao from [Odd graphs and its applications to the strong edge coloring, Applied Mathematics and Computation, 325 (2018), 246-251] in this case.

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(1,0)-Relaxed strong edge list coloring of planar graphs with girth 6

Discrete Mathematics Algorithms and Applications ◽

10.1142/s1793830919500642 ◽

2019 ◽

Vol 11 (06) ◽

pp. 1950064

Author(s):

Kai Lin ◽

Min Chen ◽

Dong Chen

Keyword(s):

Planar Graph ◽

Planar Graphs ◽

Edge Coloring ◽

List Coloring ◽

Chromatic Index ◽

Strong Chromatic Index ◽

Minimum Number ◽

Number Formula ◽

Edge List

Let [Formula: see text] be a graph. An [Formula: see text]-relaxed strong edge [Formula: see text]-coloring is a mapping [Formula: see text] such that for any edge [Formula: see text], there are at most [Formula: see text] edges adjacent to [Formula: see text] and [Formula: see text] edges which are distance two apart from [Formula: see text] assigned the same color as [Formula: see text]. The [Formula: see text]-relaxed strong chromatic index, denoted by [Formula: see text], is the minimum number [Formula: see text] of an [Formula: see text]-relaxed strong [Formula: see text]-edge-coloring admitted by [Formula: see text]. [Formula: see text] is called [Formula: see text]-relaxed strong edge [Formula: see text]-colorable if for a given list assignment [Formula: see text], there exists an [Formula: see text]-relaxed strong edge coloring [Formula: see text] of [Formula: see text] such that [Formula: see text] for all [Formula: see text]. If [Formula: see text] is [Formula: see text]-relaxed strong edge [Formula: see text]-colorable for any list assignment with [Formula: see text] for all [Formula: see text], then [Formula: see text] is said to be [Formula: see text]-relaxed strong edge [Formula: see text]-choosable. The [Formula: see text]-relaxed strong list chromatic index, denoted by [Formula: see text], is defined to be the smallest integer [Formula: see text] such that [Formula: see text] is [Formula: see text]-relaxed strong edge [Formula: see text]-choosable. In this paper, we prove that every planar graph [Formula: see text] with girth 6 satisfies that [Formula: see text]. This strengthens a result which says that every planar graph [Formula: see text] with girth 7 and [Formula: see text] satisfies that [Formula: see text].

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The strong chromatic index of (3,Δ)-bipartite graphs

Discrete Mathematics ◽

10.1016/j.disc.2016.10.016 ◽

2017 ◽

Vol 340 (5) ◽

pp. 1143-1149 ◽

Cited By ~ 5

Author(s):

Mingfang Huang ◽

Gexin Yu ◽

Xiangqian Zhou

Keyword(s):

Bipartite Graphs ◽

Chromatic Index ◽

Strong Chromatic Index

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Strong chromatic index of products of graphs

Discrete Mathematics & Theoretical Computer Science ◽

10.46298/dmtcs.414 ◽

2007 ◽

Vol Vol. 9 no. 1 (Graph and Algorithms) ◽

Author(s):

Olivier Togni

Keyword(s):

Cartesian Product ◽

Complete Graphs ◽

Chromatic Index ◽

Induced Matching ◽

Colour Class ◽

Strong Chromatic Index ◽

Minimum Number ◽

International Audience

Graphs and Algorithms International audience The strong chromatic index of a graph is the minimum number of colours needed to colour the edges in such a way that each colour class is an induced matching. In this paper, we present bounds for strong chromatic index of three different products of graphs in term of the strong chromatic index of each factor. For the cartesian product of paths, cycles or complete graphs, we derive sharper results. In particular, strong chromatic indices of d-dimensional grids and of some toroidal grids are given along with approximate results on the strong chromatic index of generalized hypercubes.

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On the strong chromatic index and maximum induced matching of tree-cographs, permutation graphs and chordal bipartite graphs

Journal of Discrete Algorithms ◽

10.1016/j.jda.2014.11.004 ◽

2015 ◽

Vol 30 ◽

pp. 21-28 ◽

Cited By ~ 1

Author(s):

Ton Kloks ◽

Sheung-Hung Poon ◽

Chin-Ting Ung ◽

Yue-Li Wang

Keyword(s):

Bipartite Graphs ◽

Chromatic Index ◽

Permutation Graphs ◽

Induced Matching ◽

Strong Chromatic Index ◽

Maximum Induced Matching ◽

Chordal Bipartite Graphs

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

Discrete Mathematics Algorithms and Applications ◽

10.1142/s1793830920500354 ◽

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.

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