Strong edge-coloring for planar graphs with large girth

Coloring, List Coloring, and Painting Squares of Graphs (and Other Related Problems)

“…The best progress on Conjecture 36 is due to Chen, Deng, Yu, and Zhou [51], who proved the following. Let G be connected and planar with girth g, where g ⩾ 26.…”

Coloring, List Coloring, and Painting Squares of Graphs (and other related problems)

“…The best progress on Conjecture 4.11 is due to Chen, Deng, Yu, and Zhou [48], who proved the following. Let G be a connected planar with girth g, where g ≥ 26.…”

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.…”

“…Conjecture 28 (Hocquard, Lajou, and Lužar [15]). Every 3-edge-colorable subcubic graph is (1, 2 ) 6 -packing edge colorable.…”

“…They showed that there are subcubic graphs that are not (1, 2, 2, 2, 2, 2, 2)-packing (abbreviated to (1, 2 ) 6 packing) edge-colorable and asked the question whether every subcubic graph is (1,2 ) 7 -packing edgecolorable. 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.…”