Abstract
Graphs and Algorithms Giving a planar graph G, let χ'l(G) and χ''l(G) denote the list edge chromatic number and list total chromatic number of G respectively. It is proved that if G is a planar graph without non-induced 7-cycles, then χ'l(G)≤Δ(G)+1 and χ''l(G)≤Δ(G)+2 where Δ(G)≥7.
Highlights
The terminology and notation used but undefined in this paper can be found in [1]
If u1, u2, · · ·, un are the vertices on the boundary walk of a face f, we write f = u1u2 · · · un
Part (b) of Conjecture 2 was proved by Hou et al for planar graphs G with ∆(G) ≥ 9 [10]
Summary
List Edge and List Total Colorings of Planar Graphs without non-induced 7-cycles†. Giving a planar graph G, let χl(G) and χl (G) denote the list edge chromatic number and list total chromatic number of G respectively. It is proved that if G is a planar graph without non-induced 7-cycles, χl(G) ≤ ∆(G) + 1 and χl (G) ≤ ∆(G) + 2 where ∆(G) ≥ 7
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