Note
Characterizations of graphs G having all [1,k]-factors in kG

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Abstract

Let k1 be an integer and G be a graph. Let kG denote the graph obtained from G by replacing each edge of G with k parallel edges. We say that G has all [1,k]-factors or all fractional [1,k]-factors if G has an h-factor or a fractional h-factor for every function h:V(G){1,2,,k} with h(V(G)) even. In this note, we come up with simple characterizations of a graph G such that kG has all [1,k]-factors or all fractional [1,k]-factors. These characterizations are extensions of Tutte’s 1-Factor Theorem and Tutte’s Fractional 1-Factor Theorem.

Keywords

All (g,f)-factors
All fractional factors
Multigraph
Isolated vertices
Tutte-type theorem

Cited by (0)

1

Supported by the National Natural Science Foundation of China under Grant​ No.11871391 and China Postdoctoral Science Foundation Funded Project.

2

Supported by JSPS KAKENHI Grant Number 19K03597.

3

Supported by the Discovery Grant of NSERC, and Shaanxi Hundred-Talent Program .