Abstract
A total weighting of a graph G is a mapping \(\varphi \) that assigns a weight to each vertex and each edge of G. The vertex-sum of \(v \in V(G)\) with respect to \(\varphi \) is \(S_{\varphi }(v)=\sum _{e\in E(v)}\varphi (e)+\varphi (v)\). A total weighting is proper if adjacent vertices have distinct vertex-sums. A graph \(G=(V,E)\) is called \((k,k{^{\prime }})\)-choosable if the following is true: For any total list assignment L which assigns to each vertex v a set L(v) of k real numbers, and assigns to each edge e a set L(e) of \(k{^{\prime }}\) real numbers, there is a proper total weighting \(\varphi \) with \(\varphi (y)\in L(y)\) for any \(y \in V \cup E\). In this paper, we prove that for any graph \(G\ne K_1\), for any positive integer m, the m-cone graph of G is (1, 4)-choosable. Moreover, we give some sufficient conditions for the m-cone graph of G to be (1, 3)-choosable. In particular, if G is a tree, a complete bipartite graph or a generalized \(\theta \)-graph, then the m-cone graph of G is (1, 3)-choosable.
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T.-L. Wong: Grant Numbers: NSC102-2115-M-110-006-MY2.
X. Zhu: Grant Numbers: NSF11171310.
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Tang, Y., Wong, TL. & Zhu, X. Total Weight Choosability of Cone Graphs. Graphs and Combinatorics 32, 1203–1216 (2016). https://doi.org/10.1007/s00373-015-1633-3
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DOI: https://doi.org/10.1007/s00373-015-1633-3