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Total weight choosability of graphs

✍ Scribed by Tsai-Lien Wong; Xuding Zhu

Publisher: John Wiley and Sons
Year: 2010
Tongue: English
Weight: 152 KB
Volume: 66
Category: Article
ISSN: 0364-9024
DOI: 10.1002/jgt.20500

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✦ Synopsis

A graph G = (V , E) is called (k, k )-total weight choosable if the following holds: For any total list assignment L which assigns to each vertex x a set L(x) of k real numbers, and assigns to each edge e a set L(e) of k real numbers, there is a mapping f : V ∪E → R such that f (y) ∈ L(y) for any y ∈ V ∪E and for any two adjacent vertices x, x , e∈E(x) f (e)+f (x) = e∈E(x ) f (e)+f (x ). We conjecture that every graph is (2, 2)-total weight choosable and every graph without isolated edges is (1, 3)-total weight choosable. It follows from results in [7] that complete graphs, complete bipartite graphs, trees other than K 2 are (1, 3)-total weight choosable. Also a graph G obtained from an arbitrary graph H by subdividing each edge with at least three vertices is (1, 3)-total weight choosable. This article proves that complete graphs, trees, generalized theta graphs are (2, 2)-total weight choosable. We also prove that for any graph H, a graph G obtained from H by subdividing each edge with at least two vertices is (2, 2)-total weight

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