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Odd subgraphs and matchings

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Abstract

Let G be a graph and f:V(G)→{1,3,5,…}. Then a subgraph H of G is called a (1,f)-odd subgraph if degH(x)∈{1,3,…,f(x)} for all x∈V(H). If f(x)=1 for all x∈V(G), then a (1,f)-odd subgraph is nothing but a matching. A (1,f)-odd subgraph H of G is said to be maximum if G has no (1,f)-odd subgraph K such that |K|>|H|. We show that (1,f)-odd subgraphs have some properties similar to those of matchings, in particular, we give a formula for the order of a maximum (1,f)-odd subgraph, which is similar to that for the order of a maximum matching.

Keywords

Odd subgraph
Graph factor
Matching

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Research supported by OTKA grants OTKA T 029772 and T 030059, and The Japan Society for the Promotion of Science.