Fractional Factors, Component Factors and Isolated Vertex Conditions in Graphs

For a graph $G = (V, E)$, a fractional $[a, b]$-factor is a real valued function h:E(G)→ [0,1] that satisfies a ≤ ~ ∑e∈ EG(v) h(e) ~ ≤ b for all v∈ V(G), where a and b are real numbers and EG(v) denotes the set of edges incident with v. In this paper, we prove that the condition iso(G-S) ≤ (k+1/2)|S| is equivalent to the existence of fractional [1,k+ 1/2]-factors, where iso(G-S) denotes the number of isolated vertices in $G-S$. Using fractional factors as a tool, we construct component factors under the given isolated conditions. Namely, (i) a graph G has a ₂,C₃,P₅, T(3)\-factor if and only if iso(G-S) ≤ 3/2|S| for all S⊂ V(G); (ii) a graph G has a 1,1, K1,2, …, K1,k, T(2k+1)\-factor (k≥ 2) if and only if iso(G-S) ≤ (k+1/2)|S| for all S⊂ V(G), where T(3) and T(2k+1) are two special families of trees.