Strong edge-colorings of sparse graphs with large maximum degree

Abstract

A strong k-edge-coloring of a graph G is a mapping from E(G) to {1, 2, . . . , k} such that every two adjacent edges or two edges adjacent to the same edge receive distinct colors. The strong chromatic index χ s (G) of a graph G is the smallest integer k such that G admits a strong k-edge-coloring.

Strong edge-coloring of 2-degenerate graphs

“…This implies that χ ′ s (G) ≤ 6∆ − 7 for 2-degenerate graph G. Choi, Kim, Kostochka, and Raspaud [5] further improved it to χ ′ s (G) ≤ 5∆ + 1 in 2016. Many believe that the optimal bound should be 4∆ + C for some constant C, but no progress has yet been made.…”

Every subcubic multigraph is (1,27) $(1,{2}^{7})$‐packing edge‐colorable

“…The strong chromatic index of a graph $$G$$ is the minimum integer $$k$$ such that $$G$$ has a $$({2}^{k})$$‐packing edge‐coloring. Strong edge‐colorings were first introduced by Fouquet and Jolivet [9] and then extended by many researchers [1, 6, 7, 11, 16–21]. Therefore, one can view $$({1}^{j},{2}^{k-j})$$‐packing edge‐colorings, as an intermediate form of coloring between proper edge‐colorings and strong edge‐colorings.…”

“…Very recently, Hocquard, Lajou, and Lužar showed that every subcubic graph is (1, 2 ) 8 -packing edge-colorable and every 3-edge-colorable subcubic graph is (1,2 ) 7 -packing edge-colorable. Furthermore, they also conjectured that every subcubic graph is (1, 2 ) 7 -packing edge-colorable.…”

“…Conjecture 1 (Hocquard, Lajou, and Lužar [15]). Every subcubic graph has a (1, 2 ) 7 -packing edge-coloring.…”

List neighbor sum distinguishing edge coloring of subcubic graphs

“…The notion of strong edge-coloring was first introduced by Fouquet and Jolivet [9], and later it was conjectured by Erdős and Nešetřil [7] that the strong chromatic index of a graph G is bounded by 5 4 ∆ 2 if ∆ is even and 5 4 ∆ 2 − 1 2 ∆ + 1 4 if ∆ is odd. Many researchers have made progress toward this conjecture and it includes but not limited to [1,5,6,10,22,23].…”