Minimum choosability of planar graphs

Huijuan Wang (), Bin Liu (), Ling Gai (), Hongwei Du () and Jianliang Wu ()
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Huijuan Wang: Qingdao University
Bin Liu: Ocean University of China
Ling Gai: Shanghai University
Hongwei Du: Harbin Institute of Technology Shenzhen Graduate School
Jianliang Wu: Shandong University

Journal of Combinatorial Optimization, 2018, vol. 36, issue 1, No 2, 13-22

Abstract: Abstract The problem of list coloring of graphs appears in practical problems concerning channel or frequency assignment. In this paper, we study the minimum number of choosability of planar graphs. A graph G is edge-k-choosable if whenever every edge x is assigned with a list of at least k colors, L(x)), there exists an edge coloring $$\phi $$ ϕ such that $$\phi (x) \in L(x)$$ ϕ ( x ) ∈ L ( x ) . Similarly, A graph G is toal-k-choosable if when every element (edge or vertex) x is assigned with a list of at least k colors, L(x), there exists an total coloring $$\phi $$ ϕ such that $$\phi (x) \in L(x)$$ ϕ ( x ) ∈ L ( x ) . We proved $$\chi '_{l}(G)=\Delta $$ χ l ′ ( G ) = Δ and $$\chi ''_{l}(G)=\Delta +1$$ χ l ′ ′ ( G ) = Δ + 1 for a planar graph G with maximum degree $$\Delta \ge 8$$ Δ ≥ 8 and without chordal 6-cycles, where the list edge chromatic number $$\chi '_{l}(G)$$ χ l ′ ( G ) of G is the smallest integer k such that G is edge-k-choosable and the list total chromatic number $$\chi ''_{l}(G)$$ χ l ′ ′ ( G ) of G is the smallest integer k such that G is total-k-choosable.

Keywords: List coloring; Choosability; Planar graph; Chordal; 05C15 (search for similar items in EconPapers)
Date: 2018
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Citations: View citations in EconPapers (1)

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DOI: 10.1007/s10878-018-0280-z

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