Some Results On Hamiltonian Factors Of Graphs

All graphs considered in this paper are finite simple graphs. Let G = (V(G), E{G)) be a graph, where V(G) and E{G) denote the vertex set and edge set of G, respectively. We use d_G{v) to stand for the degree of v in G. Let A(G) and δ(G) denote the maximum and the minimum vertex degree of G. For any S (?) V(G), the remaining subgraph of G obtained by deleting all the vertices in S and all the edges incident with the vertices in S is denoted by G — 5. If S = {v},v ∈ V(G), we usually denote G-v = G - {v}. For any X (?) E(G), the remaining subgraph of G obtained by deleting all the edges of X is denoted by G - X. If X = {e},e ∈ E(G), we usually denote G -e = G - {e}. We call a graph G a bipartite graph if the vertex set of G can be partitioned into two subsets X and Y, so that each edge has one end in X and the another end in Y, such a graph is denoted by G = (X, Y, E(G)); G = (X, Y, E{G)) is called balanced if |X| = |Y|.Let g, f be two integral functions denned on V(G) and g{v) ≤ f(v), for any v ∈ V(G). Let H be a spanning subgraph of G. We call H a (g, f)-factor of G ifSimilarly, H is a [a, b]-factor of G, if a ≤ d_H{v) ≤ b, for any v ∈V(H)(=V(G)) where a and 6 are two nonnegative integers, and H is a k-factor of G if d_H (v) = k for any v∈ V(G), where k is a positive integer. A 1-factor of G is in fact a perfect matching. We call a factor of G a Hamiltonian factor if it contains a Hamiltonian cycle of G. For a graph G, its order is |V(G)|, and we often denote it by n. We call a graph n/2 -critical if δ{G) ≥n/2 and δ{G - e) < n/2 for any e ∈ E(G).Let h{e) ∈ [0,1] be a function defined on E{G). Let d_G~h {v) = where E_v = {e : e = vw ∈ E(G)}. We call d_G~h(v) the fractional degree of v in G. We call h a indictor function if g(v) ≤ d_G~h{v) ≤ f(v) for any v ∈ V(G), Let E~h = {e ∈ E(G) : h(e) ≠ 0} and G_h be a spanning subgraph of G such that E(G_h) = E~h, then we call G_h a fractional (g,f)-factor and h is the indictor...