On the Fractional Matching Extendability of Cayley Graphs of Abelian Groups

Fractional matching extendability is a concept that brings together two widely studied topics in graph theory, namely that of fractional matchings and that of matching extendability. A fractional matching of a graph Γ with edge set E is a function f from E to the real interval $[0,1]$ with the property that for each vertex v of Γ, the sum of f-values of all the edges incident to v is at most $1$. When this sum equals $1$ for each vertex v, the fractional matching is perfect. A graph of order at least $2t+1$ is fractional t-extendable if it contains a matching of size t and if each such matching M can be extended to a fractional perfect matching in the sense that the corresponding function f assigns value $1$ to each edge of M. In this paper, we study fractional matching extendability of Cayley graphs of Abelian groups. We show that, except for the odd cycles, all connected Cayley graphs of Abelian groups are fractional $1$-extendable and we classify the fractional $2$-extendable Cayley graphs of Abelian groups. This extends the classification of $2$-extendable (in the classical sense) connected Cayley graphs of Abelian groups of even order from 1995, obtained by Chan, Chen and Yu.