Edge Colorings of Planar Graphs without 6-Cycles with Two Chords

Abstract

It is proved here that if a planar graph has maximum degree at least 6 and any 6-cycle contains at most one chord, then it is of class 1.

Planar Graphs of Maximum Degree 6 and without Adjacent 8-Cycles Are 6-Edge-Colorable

“…(1) Δ ≥ 3 and g ≥ 8, or Δ ≥ 4 and g ≥ 5, or Δ ≥ 5 and g ≥ 4 ([1]) (2) Δ ≥ 6 without intersecting 3-cycles ( [4]) (3) Δ ≥ 6 without 4-or 5-cycles ( [5]) (4) Δ ≥ 6 without 5-or 6-cycles with chords [6] (5) Δ ≥ 6 without 5-cycles ( [7]) or 6-cycles ( [8]) with two chords (6) Δ ≥ 6 without 7-cycles ( [9]) or 6-cycles ( [10]) with three chords (7) Δ ≥ 6 and any vertex is incident with at most three triangles ( [11]) (8) Δ ≥ 6 and without adjacent k-cycles, where 3 ≤ k ≤ 6…”

Edge colorings of planar graphs

“…Nevertheless, many interesting results have been reached in recent years, confirming the conjecture for planar graphs with maximum degree 6 subject to various conditions and constraints. We refer the interested reader to the following sources for more details [9,20,27,43,44,49,65,72,75,83,84].…”

“…In [75], Xue and Wu proved that the planar graph G with ∆ ≥ 6 and any 6-cycle contains at most one chord is ∆-edge-colorable. We extend this result to signed graphs by applying Lemma 3.5, Lemma 3.6 and Lemma 3.8.…”