On the Strong Chromatic Index of Sparse Graphs

Date
2018-01-01
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DeOrsey, Philip
Lidicky, Bernard
Ferrara, Michael
Graber, Nathan
Hartke, Stephen
Nelsen, Luke
Sullivan, Eric
Jahanbekam, Sogol
Lidicky, Bernard
Stolee, Derrick
White, Jennifer
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Abstract

The strong chromatic index of a graph G, denoted χ′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 χ′s,ℓ(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 girth(G)≥41 then χ′s,ℓ(G)≤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 girth(G)≥30, then χ′s(G)≤5, improving a bound from the same paper.

Finally, if G is a planar graph with maximum degree at most four and girth(G)≥28, then χ′s(G)N≤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.

Description
<p>This article is published as P. DeOrsey J. Diemunsch, M. Ferrara, N. Graber, S. G. Hartke, S. Jahanbekam, B. Lidický, L. Nelsen, D. Stolee, E. Sullivan. "On the Strong Chromatic Index of Sparse Graphs." <em>Electronic Journal of Combinatorics </em>25 (2018), #P3.18.</p>
Keywords
Strong Edge Coloring, Strong Chromatic Index, Sparse Graphs
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