List edge and list total colorings of planar graphs without 4-cycles

Total coloring of embedded graphs of maximum degree at least ten

“…The upper bound remains open. TCC was verified by Rosenfeld [10] and Vijayaditya [13] for Δ(G) = 3 and by Kostochka [7][8][9] for Δ(G) 5. For planar graphs the conjecture was verified by Borodin [2] for Δ(G) 9, by Yap for Δ(G) 8 (Theorem 7.2 in [18]), and by Sanders and Zhao for Δ(G) = 7 [12].…”

“…Wang [16] proved that any planar graph G of maximum degree Δ(G) 10 has χ (G) = Δ(G) + 1. Wang and Wu [15], and Hou et al [5] independently considered planar graphs without 4-cycles and got some interesting results. Recently, Hou et al [6] considered planar graphs without i-cycles for some i ∈ {5, 6} and showed that χ (G) = Δ(G)+1 if Δ(G) 8.…”

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

“…Interestingly, the list edge chromatic number and the list total chromatic number of planar graphs with large maximum degree equals a lower bound. Hou et al (2006) proved χ l (G) = Δ and χ l (G) = Δ+1 for planar graphs with Δ ≥ 8 and without 4-cycles. Li and Xu (2011) proved this result for planar graphs with Δ ≥ 8 and no 3-cycles adjacent to 4-cycles.…”

A note on edge-choosability of planar graphs without intersecting 4-cycles

“…In 1999, Juvan et al [10] settled the case for (G) = 4. Some other special cases of Conjecture 1.2 have been confirmed such as graphs with girth at least 8 (G)(ln (G) + 1.1) [11], planar graphs with (G) ≥ 9 [12], and planar graph without some certain subgraph (see [13][14][15][16][17][18][19][20]). We call two cycles intersecting if they share at least one common vertex or edge.…”