Extremal Properties of \((1, f)\)-Odd Factors in Graphs

Abstract

Let \(G\) be a simple graph and \(f: V(G) \mapsto \{1, 3, 5, \ldots\}\) an odd integer valued function defined on \(V(G)\). A spanning subgraph \(F\) of \(G\) is called a \((1, f)\)-odd factor if \(d_F(v) \in \{1, 3, \ldots, f(v)\}\) for all \(v \in V(G)\), where \(d_F(v)\) is the degree of \(v\) in \(F\). For an odd integer \(k\), if \(f(v) = k\) for all \(v\), then a \((1, f)\)-odd factor is called a \([1, k]\)-odd factor. In this paper, the structure and properties of a graph with a unique \((1, f)\)-odd factor is investigated, and the maximum number of edges in a graph of the given order which has a unique \([1, k]\)-odd factor is determined.