List edge and list total coloring of planar graphs with maximum degree 8

Abstract

Let G be a planar graph with maximum degree Δ ≥ 8 and without chordal 5-cycles. Then χ l (G) = Δ and χ l (G) = Δ + 1.

List Edge Colorings of Planar Graphs without Adjacent 7-Cycles

“…(1) If f 3 (v) � 6 and one of its edges is not incident with any 3-face (as in Figure 1,(1 − 10), (1 − 11), (1 − 13)), then 9 , where all the neighbors v i (1 ≤ i ≤ 9) of v are in an anticlockwise order. We use f i to indicate the face which is incident with v, v i , and…”

“…So far, Conjecture 1 has only been proved to be true for a few graphs, including bipartite multigraphs [3], complete graphs of odd order [4], multicircuits [5], graphs embedded in a surface of nonnegative characteristic and Δ(G) ≥ 12 [6], and outer planar graphs [7]. For planar graphs, the readers can see [8][9][10][11][12].…”

The list coloring and list total coloring of planar graphs with maximum degree at least 7

“…By Lemma 3, we can get the following result immediately. In [19], Theorem 2 was proved for ∆(G) ≥ 8. Hence, to prove Theorem 2, we just consider the case ∆(G) = 7.…”

“…For planar graphs, the best known result is that if ∆(G) ≥ 12, a planar graph G is list edge ∆(G)-colorable and list total (∆(G) + 1)-colorable [7]. It is proved that χ ′ l (G) = ∆(G) and χ ′′ l (G) = ∆(G) + 1 for a planar graph G with ∆(G) ≥ 7 and no triangle adjacent to a C 4 [6], or with no cycle of length from 4 to k, where k ≥ 4 and (∆(G), k) ∈ {(7, 4), (6,5), (5, 8)} [10], or with ∆(G) ≥ 7 and no adjacent cycles of length at most 4 [15], or with ∆(G) ≥ 8 and no cycle of length 3 adjacent to a cycle of length 5 [15], or with ∆(G) ≥ 8 and no adjacent 4-cycles [18], or with ∆(G) ≥ 8 and no chordal 5-cycle [19].…”

Total Equitable List Coloring

“…Conjecture 5 has been verified for certain planar graphs [6,28,25,38] and multicircuits [20,21]. In [19] it was conjectured that the square of every graph is chromatic-choosable (a much stronger conjecture), but this was recently disproved by Kim and Park [18].…”