On the Strong Chromatic Index of Sparse Graphs [article]

Philip DeOrsey, Jennifer Diemunsch, Michael Ferrara, Nathan Graber, Stephen G. Hartke, Sogol Jahanbekam, Bernard Lidicky, Luke L. Nelsen, Derrick Stolee, Eric Sullivan
2015 arXiv   pre-print
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
more » ... 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) ≤ 7, improving a more general bound of Wang and Zhao from [Odd graphs and its application on the strong edge coloring, arXiv:1412.8358] in this case.
arXiv:1508.03515v1 fatcat:rpyw73c4krgunnw7kzv7vipwsy