Factors and induced subgraphs

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Abstract

In [4] Katerinis proved the following: If G - x has a 2k-factor for each x ε{lunate} V(G), then G has a 2k-factor. In this paper, we prove the following generalizations. (1) Let p be a fixed integer with 0<p<|G|, and suppose g and f are non-negative integer-valued functions on V(G) with 0≤g(x)≤f(x)≤dG(x). Suppose that each induced p-vertex subgraph H of G has a (g{divides}H, f{divides}H)-factor, i.e. a spanning subgraph F with g(x)≤dF(x)≤f(x) for all x ε{lunate} V(H). Then G has a (g, f)-factor, unless g = f and Σxε{lunate}VGf(x) is odd. When g=f≡k for some integer k, the hypothesis can be weakened by requiring the existence of a desired factor only for the "vertex complements" of connected subgraphs. (2) Let G be a connected graph and p be an integer such that 0<p<|G|. Suppose k|G| is even and G - V(P) has a k-factor for each p-vertex connected induced subgraph P. Then G has a k-factor. © 1988.

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Egawa, Y., Enomoto, H., & Saito, A. (1988). Factors and induced subgraphs. Discrete Mathematics, 68(2–3), 179–189. https://doi.org/10.1016/0012-365X(88)90111-2

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