Abstract
Let G be an arbitrary spanning subgraph of the complete graph K r+1 on r+1 vertices and K r+1 − E(G) be the graph obtained from K r+1 by deleting all edges of G. A non-increasing sequence π = (d 1, d 2, ..., d n ) of nonnegative integers is said to be potentially K r+1 − E(G)-graphic if there is a graph on n vertices that has π as its degree sequence and contains K r+1 −E(G) as a subgraph. In this paper, a characterization of π that is potentially K r+1 − E(G)-graphic is given, which is analogous to the Erd−os-Gallai characterization of graphic sequences using a system of inequalities. This is a solution to an open problem due to Lai and Hu. As a corollary, a characterization of π that is potentially K s,t -graphic can also be obtained, where K s,t is the complete bipartite graph with partite sets of size s and t. This is a solution to an open problem due to Li and Yin.
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Supported by National Natural Science Foundation of China (Grant No. 11161016)
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Yin, J.H. A Rao-type characterization for a sequence to have a realization containing an arbitrary subgraph H . Acta. Math. Sin.-English Ser. 30, 389–394 (2014). https://doi.org/10.1007/s10114-014-2732-4
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DOI: https://doi.org/10.1007/s10114-014-2732-4