On total weight choosability of graphs

Haili Pan () and Daqing Yang ()
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Haili Pan: Fuzhou University
Daqing Yang: Fuzhou University

Journal of Combinatorial Optimization, 2013, vol. 25, issue 4, No 21, 766-783

Abstract: Abstract For a graph G with vertex set V and edge set E, a (k,k′)-total list assignment L of G assigns to each vertex v a set L(v) of k real numbers as permissible weights, and assigns to each edge e a set L(e) of k′ real numbers as permissible weights. If for any (k,k′)-total list assignment L of G, there exists a mapping f:V∪E→ℝ such that f(y)∈L(y) for each y∈V∪E, and for any two adjacent vertices u and v, ∑ y∈N(u) f(uy)+f(u)≠∑ x∈N(v) f(vx)+f(v), then G is (k,k′)-total weight choosable. It is conjectured by Wong and Zhu that every graph is (2,2)-total weight choosable, and every graph with no isolated edges is (1,3)-total weight choosable. In this paper, it is proven that a graph G obtained from any loopless graph H by subdividing each edge with at least one vertex is (1,3)-total weight choosable and (2,2)-total weight choosable. It is shown that s-degenerate graphs (with s≥2) are (1,2s)-total weight choosable. Hence planar graphs are (1,10)-total weight choosable, and outerplanar graphs are (1,4)-total weight choosable. We also give a combinatorial proof that wheels are (2,2)-total weight choosable, as well as (1,3)-total weight choosable.

Keywords: Total weighting; Edge weighting; Vertex coloring (search for similar items in EconPapers)
Date: 2013
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DOI: 10.1007/s10878-012-9491-x

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