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Pure and Applied Mathematics Quarterly
Volume 18 (2022)
Number 6
Special issue in honor of Fan Chung
Guest editors: Paul Horn, Yong Lin, and Linyuan Lu
An alternative proof of general factor structure theorem
Pages: 2551 – 2568
DOI: https://dx.doi.org/10.4310/PAMQ.2022.v18.n6.a10
Authors
Abstract
Let $G$ be a graph, and $H : V (G) \to 2^\mathbb{N}$ a set function associated with $G$. A spanning subgraph $F$ of graph $G$ is called a general factor or an $H$-factor of $G$ if $d_F (x) \in H(x)$ for every vertex $x \in V (G)$. The existence of $H$-factors is, in general, an $NP$-complete problem. $H$-factor problems are considered as one of most general factor problem because many well-studied factors (e.g., perfect matchings, $f$-factor problems and $(g,f)$-factor problems) are special cases of $H$-factors. Lovász [“The factorization of graphs (II),” Acta Math. Hungar., 23 (1972), 223–246] gave a structure description of $H$-optimal subgraphs and obtained a deficiency formula. In this paper, we introduce a new type of alternating path to study Lovász’s canonical structural partition of graphs and consequently obtain an alternative and shorter proof of Lovász’s deficiency formula for $H$-factors. Moreover, we also obtain new properties regarding Lovász’s canonical structural partition of $H$-factors.
Keywords
degree constrained factor, alternating path, changeable trail
2010 Mathematics Subject Classification
05C70
This work is supported by the National Natural Science Foundation of China (No. 12271425), and Natural Sciences and Engineering Research Council of Canada (RGPIN-2019-06429).
Received 25 May 2021
Received revised 22 June 2022
Accepted 21 July 2022
Published 29 March 2023