For the list object, introduced in Chapter 5, it was shown that each data element contains at most one predecessor element and one successor element. Therefore, for any given data element or node in the list structure, we can talk in terms of a next element and a previous element.

Graphs and Digraphs

A graph G is called a fractional (g, f, m)-deleted graph if the resulting graph admits a fractional (g, f)-factor after m edges are removed. An important fact in the characterization of a discrete graph dynamical system is played by the independent sets, i.e., subsets of the vertex set of G in which any two of them are not adjacent because they reflect the sparsity and stability of the graph system in somehow. The neighborhoods union of independent sets characterizes the local density and local clustering characteristics of the graph system. In this paper, we study the relationship between characteristics of independent sets and fractional (g, f, m)-deleted graph systems. The main contributions cover two aspects: first, we present an independent set degree condition for a graph to be fractional (g, f, m)-deleted; later an independent set neighborhood union condition for fractional (g, f, m)-deleted graphs is determined. Furthermore, we show that the results obtained are the best in some sense.

Two Tight Independent Set Conditions for Fractional ( g , f , m )-Deleted Graphs Systems

For a set ℋ of connected graphs, a spanning subgraph H of a graph G is called an ℋ-factor of G if every component of H is isomorphic to a member ofℋ An H -factor is also referred as a component factor If each component of H is a star (resp path), H is called a star (resp path) factor By a P ≥ k -factor (k positive integer) we mean a path factor in which each component path has at least k vertices (ie it has length at least k  − 1) A graph G is called a P ≥ k -factor covered graph, if for each edge e of G , there is a P ≥ k -factor covering e  In this paper, we prove that (1) a graph G has a {K 1,1 ,K 1,2 , … ,K 1,k }-factor if and only if bind(G ) ≥ 1/k , where k  ≥ 2 is an integer; (2) a connected graph G is a P ≥ 2 -factor covered graph if bind(G ) > 2/3; (3) a connected graph G is a P ≥ 3 -factor covered graph if bind(G ) ≥ 3/2 Furthermore, it is shown that the results in this paper are best possible in some sense

Some results about component factors in graphs

A graph G is called a fractional (g, f)-covered graph if for any e ∈ E(G), G admits a fractional (g, f)-factor covering e. A graph G is called a fractional (g, f, n)-critical covered graph if for any S ⊆ V(G) with ∣S∣ = n, G − S is a fractional (g, f)-covered graph. A fractional (g, f, n)-critical covered graph is said to be a fractional (a, b, n)-critical covered graph if g(x) = a and f(x) = b for every x ∈ V(G). A fractional (a, b, n)-critical covered graph was first defined and studied in [1]. In this article, we investigate fractional (g, f, n)-critical covered graphs and present a binding number condition for the existence of fractional (g, f, n)-critical covered graphs, which is an improvement and generalization of a previous result obtained in [2].

Research on Fractional Critical Covered Graphs

A spanning subgraph of a graph is defined as a path factor of the graph if its component are paths. A P ≥n -factor means a path factor with each component having at least n vertices. A graph G is defined as a (P ≥n , m )-factor deleted graph if G –E ′ has a P ≥n -factor for every E ′ ⊆ E (G ) with |E ′| = m . A graph G is defined as a (P ≥n , k )-factor critical graph if after deleting any k vertices of G the remaining graph of G admits a P ≥n -factor. In this paper, we demonstrate that (i) a graph G is (P ≥3 , m )-factor deleted if κ (G ) ≥ 2m  + 1 and bind (G ) ≥  2/3 - ; (ii) a graph G is (P ≥3 , k )-factor critical if κ (G ) ≥ k  + 2 and bind (G ) ≥ .

Remarks on path factors in graphs

Let D be a digraph with vertex set V(D) and arc set A(D) and let f=(f−, f+) be a pair of functions defined on V(D). Let H be a km-subdigraph of D. In this paper, it is proved that every (0, mf−m+1)-digraph has a (0, f)-factorization k-orthogonal to H if f(x)≥3k−2 for each x∈V(D).

Remarks on orthogonal factorizations of digraphs

Let G be a graph, and let a, b, k be integers with 0≤a≤b, k≥0. An [a, b]-factor of graph G is defined as a spanning subgraph F of G such that a≤dF(x)≤b for each x∈V(G). Then a graph G is called an (a, b, k)-critical graph if after deleting any k vertices of G the remaining graph of G has an [a, b]-factor. In this article, a sufficient condition is given, which is a neighborhood condition for a graph G to be an (a, b, k)-critical graph.

A sufficient condition for a graph to be an (a, b, k)-critical graph

Let G be a graph, and k a positive integer Let h: E(G)→[0, 1] be a function If ∑e∋ xh(e)=k holds for any x∈V(G), then we call G[Fh] a fractional k-factor of G with indicator function h, where Fh={e∈E(G):h(e)>0} In this paper, we prove some results on the existence of a fractional k-factor in a simple graph depending on δ(G) and α(G) Furthermore, we show that the results are best possible in some sense

Sizhong Zhou

Papers

Remarks on orthogonal factorizations of digraphs

A sufficient condition for a graph to be an (a, b, k)-critical graph

Some new sufficient conditions for graphs to have fractional k-factors