Planar graphs without cycles of lengths 4 and 5 and close triangles are DP-3-colorable

Abstract

Montassier, Raspaud, and Wang (2006) asked to find the smallest positive integers d 0 and d 1 such that planar graphs without {4, 5}-cycles and d ∆ ≥ d 0 are 3-choosable and planar graphs without {4, 5, 6}-cycles and d ∆ ≥ d 1 are 3-choosable, where d ∆ is the smallest distance between triangles. They showed that 2 ≤ d 0 ≤ 4 and d 1 ≤ 3. In this paper, we show that the following planar graphs are DP-3-colorable: (1) planar graphs without {4, 5}-cycles and d ∆ ≥ 3 are DP-3-colorable, and (2) planar graphs without {4, 5, 6}-cycles and d ∆ ≥ 2 are DP-3-colorable. DP-coloring is a generalization of list-coloring, thus as a corollary, d 0 ≤ 3 and d 1 ≤ 2. We actually prove stronger statements that each pre-coloring on some cycles can be extended to the whole graph.

Planar Graphs with the Distance of 6--Cycles at Least 2 from Each Other Are DP-3-Colorable

“…DP-3-colorable planar graphs can be found in [17,18] and DP-4-colorable planar graphs can be found in [19][20][21]. Yin and Yu [22] proved planar graphs with no {4, 5, 6}-cycles in which the distance between triangles is at least 2 are DP-3-colorable. We present the following result in this paper.…”

Planar graphs having no cycle of length 4, 7, or 9 are DP‐3‐colorable

“…Denote by △ d the smallest distance between triangles. The DP-3-colorability was proved for planar graphs with ≥ △ d 3 and with no cycle of length from {4, 5} [17], planar graphs with ≥ △ d 2 and with no cycle of length from {4, 5, 6} [17], {4, 5, 7} [13], {5, 6, 7} [10], or {5, 6, 8} [13], planar graphs with neither intersecting triangles nor cycle of length from {4, 5, 6, 7} [12], planar graphs with neither adjacent triangles nor cycle of length from {5, 6, 9} [13], and planar graphs with no cycle of length from {3, 5, 6} [10], {3, 6, 7, 8} [10], {4, 5, 6, 9} [10], {4, 5, 7, 9} [10], {4, 6, 7, 9} [9], {4, 6, 8, 9} [9], or {4, 7, 8, 9} [9].…”

DP-3-coloring of planar graphs without certain cycles

“…In general, we use dist to denote the minimum distance of two triangles in a graph. Yin and Yu [27] gave the following Bordeaux condition for planar graphs to be DP-3-colorable.…”