# Minimum choosability of planar graphs

Minimum choosability of planar graphs 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. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Journal of Combinatorial Optimization Springer Journals

# Minimum choosability of planar graphs

, Volume 36 (1) – Mar 23, 2018
10 pages

/lp/springer_journal/minimum-choosability-of-planar-graphs-p0RjsuFezz
Publisher
Springer US
Subject
Mathematics; Combinatorics; Convex and Discrete Geometry; Mathematical Modeling and Industrial Mathematics; Theory of Computation; Optimization; Operations Research/Decision Theory
ISSN
1382-6905
eISSN
1573-2886
D.O.I.
10.1007/s10878-018-0280-z
Publisher site
See Article on Publisher Site

### 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.

### Journal

Journal of Combinatorial OptimizationSpringer Journals

Published: Mar 23, 2018

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