All fractional (g,f)-factors in graphs

zbMath1463.05454arXiv1412.3882MaRDI QIDQ4969548

Si-zhong Zhou, Zhi-ren Sun

Publication date: 13 October 2020

Abstract: Let $G$ be a graph, and $g,f:V(G)ightarrowN$ be two functions with $g(x)leqf(x)$ for each vertex $x$ in $G$. We say that $G$ has all fractional $(g,f)$-factors if $G$ includes a fractional $r$-factor for every $r:V(G)ightarrowN$ such that $g(x)leqr(x)leqf(x)$ for each vertex $x$ in $G$. Let $H$ be a subgraph of $G$. We say that $G$ admits all fractional $(g,f)$-factors including $H$ if for every $r:V(G)ightarrowN$ with $g(x)leqr(x)leqf(x)$ for each vertex $x$ in $G$, $G$ includes a fractional $r$-factor $F_h$ with $h(e)=1$ for any $einE(H)$, then we say that $G$ admits all fractional $(g,f)$-factors including $H$, where $h:E(G)ightarrow[0,1]$ is the indicator function of $F_h$. In this paper, we obtain a characterization for the existence of all fractional $(g,f)$-factors including $H$ and pose a sufficient condition for a graph to have all fractional $(g,f)$-factors including $H$.

Full work available at URL: https://arxiv.org/abs/1412.3882

zbMATH Keywords

fractional \((g,f)\)-factorall fractional \((g,f)\)-factors

Mathematics Subject Classification ID

Edge subsets with special properties (factorization, matching, partitioning, covering and packing, etc.) (05C70) Fractional graph theory, fuzzy graph theory (05C72)

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