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• W2023179859 abstract "Let G be a planar graph with no two 3-cycles sharing an edge. We show that if �(G) � 9, then � '(G) = �(G) and � '' (G) = �(G) + 1. We also show that if �(G) � 6, then � '(G) � �(G) + 1 and if �(G) � 7, then � '' (G) � �(G) + 2. All of these results extend to graphs in the projective plane and when �(G) � 7 the results also extend to graphs in the torus and Klein bottle. This second edge-choosability result improves on work of Wang and Lih and of Zhang and Wu. All of our results use the discharging method to prove structural lemmas about the existence of subgraphs with small degree-sum. For example, we prove that if G is a planar graph with no two 3-cycles sharing an edge and with �(G) � 7, then G has an edge uv with d(u) � 4 and d(u) + d(v) � �(G) + 2. All of our proofs yield linear-time algorithms that produce the desired colorings. MSC: 05C15, 05C10 All our graphs are finite and without loops or multiple edges. Let G be a plane graph. We use E(G), V (G), F (G), �( G), and �(G) to denote the edge set, vertex set, face set, maximum degree, and minimum degree of G, respectively. When the graph is clear from context, we use � , rather than �( G). We use j-face and j-vertex to mean faces and vertices of degree j. The degree of a face f is the number of edges along the boundary of f, with each cut-edge being counted twice. The degree of a face f and the degree of a vertex v are denoted by d(f) and d(v). We say a face f or vertex v is large when d(f) ≥ 5 or d(v) ≥ 5. We use triangle to mean 3-cycle. We use kite to mean a subgraph of G formed by two 3-cycles that share an edge. We use element to mean vertex or face. A proper total-coloring of G is an assignment of a label to each element so that no two incident or adjacent elements receive the same label. We call these labels colors. A proper k-total-coloring is a proper total-coloring that uses no more than k colors. A total assignment L is a function on E(G) ∪ V (G) that assigns each element x a list L(x) of colors available for use on that element. An L-total-coloring is a proper total-coloring with the additional constraint that each element receives a color appearing in its assigned list. We say that a graph G is k-total-choosable if G has a proper L-total-coloring whenever |L(x)| ≥ k for every x ∈ E(G) ∪ V (G). The total chromatic number of G, denoted � '' (G), is the least integer k such that G is k-total-colorable. The list total chromatic number of G, denoted � '' (G), is the least integer k such that G is k-total-choosable. In particular, note that � '' (G) ≤ � '' (G). The list edge chromatic number � '(G) is defined similarly in terms of coloring only edges; the ordinary edge chromatic number is denoted � ' (G). Probably the most fundamental and important result about the edge chromatic number of graphs is:" @default.
• W2023179859 created "2016-06-24" @default.
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• W2023179859 date "2009-01-01" @default.
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• W2023179859 title "Edge-choosability and total-choosability of planar graphs with no adjacent 3-cycles" @default.
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• W2023179859 doi "https://doi.org/10.7151/dmgt.1438" @default.
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