On the Strong Chromatic Index of Sparse Graphs

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 ′ ℓ,s (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 ′ ℓ,s (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.

Strong edge colorings of graphs and the covers of Kneser graphs

“…Our computational experiments show that all cubic graphs of girth at least 5 on at most 26 vertices and all bridgeless subcubic graphs of girth at least 5 on at most 18 vertices admit a strong 7-edge-coloring. We therefore propose the following conjecture, which strengthens Case (5) of Conjecture 1.2. Conjecture 5.2.…”

“…Case $$(4)$$ was established by Wu and Lin [19]; it easily follows also from a result of Maydanskiy [16]. Up to our best knowledge, Cases $$(5)$$ and $$(6)$$ are still open, although several partial results confirming Case $$(6)$$ are known [4,5].…”

Strong edge colorings of graphs and the covers of Kneser graphs

“…Case (4) was established by Wu and Lin [18]; it easily follows also from a result of Maydanskiy [15]. Up to our best knowledge, Cases (5) and (6) are still open, although several partial results confirming Case (6) are known [3,4].…”

Odd graph and its applications to the strong edge coloring

“…For n = 3, we have the following result. 9 ], there exists a 9-special walk from w 0 to w 9 such that its first edge is labeled with λ 1 and the last edge is labeled with λ 2 .…”