Closure and forbidden pairs for 2-factors

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Abstract

Pairs of connected graphs X, Y such that a graph G is 2-connected and X Y-free implies that G is hamiltonian were characterized by Bedrossian. Using the closure concept for claw-free graphs, the first author simplified the characterization by showing that if considering the closure of G, the list in the Bedrossian characterization can be reduced to one pair, namely, K1, 3, N1, 1, 1 (where Ki, j is the complete bipartite graph, and Ni, j, k is the graph obtained by identifying endvertices of three disjoint paths of lengths i, j, k to the vertices of a triangle). Faudree et al. characterized pairs X, Y such that G is 2-connected and X Y-free implies that G has a 2-factor. Recently, the first author et al. strengthened the closure concept for claw-free graphs such that the closure of a graph has stronger properties while still preserving the (non)-existence of a 2-factor. In this paper we show that, using the 2-factor closure, the list of forbidden pairs for 2-factors can be reduced to two pairs, namely, K1, 4, P4 and K1, 3, N1, 1, 3. © 2010 Elsevier B.V. All rights reserved.

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Ryjáček, Z., & Saburov, K. (2010). Closure and forbidden pairs for 2-factors. Discrete Mathematics, 310(10–11), 1564–1572. https://doi.org/10.1016/j.disc.2010.02.006

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