List Edge and List Total Colourings of Multigraphs

The structure of plane graphs with independent crossings and its applications to coloring problems

“…As far as we know, this conjecture was confirmed for several classes of graphs including planar graphs with maximum degree at least 12 [6] and 1-planar graphs with maximum degree at least 21 [23]. We now focus on IC-planar graphs.…”

“…The proof idea of the claim is borrowed from the proof of [6,Theorem 8]. Actually, one can also find that the proof of it is just a part of the proof of [19, Lemma 2.4], so we omit the detailed proof of the claim here.…”

List Edge and List Total Colorings of Planar Graphs Without 6-Cycles With Chord

“…It is well known as the List Coloring Conjecture. Part (b) was formulated by Borodin, Kostochka and Woodall [2]. Part (a) and Part (b) has been proved for outerplanar graphs [13], and graphs with ∆ ≥ 12 which can be embedded in a surface of nonnegative characteristic [2].…”

“…Part (b) was formulated by Borodin, Kostochka and Woodall [2]. Part (a) and Part (b) has been proved for outerplanar graphs [13], and graphs with ∆ ≥ 12 which can be embedded in a surface of nonnegative characteristic [2]. List Coloring Conjecture has been proved for a few other special graphs, such as bipartite multigraphs [4], complete graphs of odd order [6].…”

An Improved Sufficient Condition for Reconfiguration of List Edge-Colorings in a Tree

“…In addition, they gave an infinite family of instances on paths that satisfy their sufficient condition and whose transformation requires Ω(n 2 ) intermediate L-edge-colorings. As the authors mentioned in [9], their sufficient condition is motivated by the well-known "list coloring conjecture" [11]: it is conjectured that any graph G has an L-edgecoloring if |L(e)| ≥ χ (G) for each edge e, where χ (G) is the chromatic index of G, that is, the minimum number of colors required for an ordinary edge-coloring of G. This conjecture has not been proved yet, but Borodin et al [2] proved that any bipartite graph, and hence any tree, has an L-edgecoloring if |L(e)| ≥ max{d(v), d(w)} for each edge e = vw. In this sense, there is a gap between the two sufficient conditions [2] and [9]: from the sufficient condition of [9] we cannot say anything about the reconfiguration if a given tree T has an edge e = vw with |L(e)| = max{d(v), d(w)}, whereas T has L-edge-colorings.…”