ON THE MINIMUM COVERING NUMBER FORMULA OF GRAPHS

Throughout this paper,D=（d1,d2,...,dn）denote a sequence of nonnegative inte-gers.We let（?）（D）denote the class of all graphs with degree sequence D,or equally,theclass of all symmetric（0,1）--matrices with trace 0 and row sum vector D.The structure matrix S=S（D） of D is a matrix of order n+1,whose entries

SOME CONDITIONS FOR f-COVERED GRAPHS

A graph G is f-covered if each edge of G belongs to an f-factor. Some sufficient conditions for a graph to be f-covered are given.Katerinis'and Bermond's results are generalized.

OnFractional(g,f,n′,m)-Critical Covered Graphs

The main contribution in this article is threefold:(1)we show the necessary and sufficient condition for graphs to be fractional(g,f)-covered which can be expressed in different forms,and extended to fractional(g,f,m)-covered graphs;(2)the concept of fractional-critical covered graph is put forward and its necessary and sufficient condition is given;(3)we present the degree condition for a graph to be fractional(g,f,n′,m)-critical covered,and show that degree bound is sharp when m is small.Moreover,the related result in fractional((a,b,n′,m)-critical covered setting is also verified.

Circulant Double Coverings of a Circulant Graph of Valency Five

Enumerating the isomorphism classes of several types of graph covering projections is one of the central research topics in enumerative topological graph theory. A covering of G is called circulant if its covering graph is circulant. Recently, the authors [Discrete Math., 277, 73-85 （2004）1 enumerated the isomorphism classes of circulant double coverings of a certain type, called a typical covering, and showed that no double covering of a circulant graph of valency three is circulant. Also, in [Graphs and Combinatorics, 21,386 400 （2005）], the isomorphism classes of circulant double coverings of a circulant graph of valency four are enumerated. In this paper, the isomorphism classes of circulant double coverings of a circulant graph of valency five are enumerated.

Enumerating Abelian Typical Cube-Free Fold Coverings of a Circulant Graph

Enumerating the isomorphism or equivalence classes of several types of graph coverings is one of the central research topics in enumerative topological graph theory.A covering projection p from a Cayley graph Cay(Г,X)onto another Cayley graph Cay(Q,y)is called typical if the function p:Г→Q on the vertex sets is a group epimorphism.A typical covering is called abelian(or circulant,respectively)if its covering graph is a Cayley graph on an abelism(or a cyclic,respectively)group.Recently,the equivalence classes of connected abelian typical prime-fold coverings of a circulant graph are enumerated.As a continuation of this work,we enumerate the equivalence classes of connected abelian typical cube-free fold coverings of a circulant graph.

Path Factors and Neighborhoods of Independent Sets in Graphs

A path-factor is a spanning subgraph F of G such that every component of F is a path with at least two vertices.Let k≥2 be an integer.A P_(≥k)-factor of G means a path factor in which each component is a path with at least k vertices.A graph G is a P_(≥k)-factor covered graph if for any e∈E(G),G has a P_(≥k)-factor including e.Letβbe a real number with 1/3≤β≤1 and k be a positive integer.We verify that(ⅰ)a k-connected graph G of order n with n≥5k+2 has a P_(≥3)-factor if|NG(I)|>β(n-3k-1)+k for every independent set I of G with|I|=「β(2k+1)」;(ⅱ)a(k+1)-connected graph G of order n with n≥5k+2 is a P_(≥3)-factor covered graph if|NG(I)|>β(n-3k-1)+k+1 for every independent set I of G with|I|=「β(2k+1)」.

Semisymmetric Cubic Graphs as Regular Covers of K_(3,3)

A regular graph X is called semisymmetric if it is edge-transitive but not vertex-transitive. For G ≤ AutX, we call a G-cover X semisymmetric if X is semisymmetric, and call a G-cover X one-regular if Aut X acts regularly on its arc-set. In this paper, we give the sufficient and necessary conditions for the existence of one-regular or semisymmetric Zn-Covers of K3,3. Also, an infinite family of semisymmetric Zn×Zn-covers of K3,3 are constructed.

Some Existence Theorems on Path Factors with Given Properties in Graphs

A path factor of G is a spanning subgraph of G such that its each component is a path.A path factor is called a P≥n-factor if its each component admits at least n vertices.A graph G is called P≥n-factor covered if G admits a P≥n-factor containing e for any e∈E(G),which is defined by[Discrete Mathematics,309,2067-2076(2009)].We first define the concept of a(P≥n,k)-factor-critical covered graph,namely,a graph G is called(P≥n,k)-factor-critical covered if G-D is P≥n-factor covered for any D⊆V(G)with|D|=k.In this paper,we verify that(i)a graph G withκ(G)≥k+1 is(P≥2,k)-factor-critical covered if bind(G)>2+k/3;(ii)a graph G with|V(G)|≥k+3 andκ(G)≥k+1 is(P≥3,k)-factor-critical covered if bind(G)≥4+k/3.

A Result on Fractional(a,b,k)-critical Covered Graphs

A fractional[a,b]-factor of a graph G is a function h from E(G)to[0,1]satisfying a≤d^(h)_(G)(v)≤b for every vertex v of G,where d^(h)_(G)(v)=∑e∈E(v)h(e)and E(v)={e=uv:u∈V(G)}.A graph G is called fractional[a,b]-covered if G contains a fractional[a,b]-factor h with h(e)=1 for any edge e of G.A graph G is called fractional(a,b,k)-critical covered if G—Q is fractional[a,b]-covered for any Q⊆V(G)with∣Q∣=k.In this article,we demonstrate a neighborhood condition for a graph to be fractional(a,b,k)-critical covered.Furthermore,we claim that the result is sharp.

Bipartite double cover and perfect 2-matching covered graph with its algorithm

Let B（G） denote the bipartite double cover of a non-bipartite graph G with v ≥ 2 vertices and s edges. We prove that G is a perfect 2-matching covered graph if and only if B（G） is a 1-extendable graph. Furthermore, we prove that B（G） is a minimally l-extendable graph if and only if G is a minimally perfect 2-matching covered graph and for each e = xy ∈ E（G）, there is an independent set S in G such that ｜ГG（S）｜ = ｜S｜ ＋ 1, x ∈ S and ｜ГG-xy（S）｜ = ｜S｜ Then, we construct a digraph D from B（G） or G and show that D is a strongly connected digraph if and only if G is a perfect 2-matching covered graph. So we design an algorithm in O（x√vε） time that determines whether G is a perfect 2-matching covered graph or not.

Discussion on Fractional(a,b,k)-critical Covered Graphs

A graph G is called a fractional[a,b]-covered graph if for each e∈E(G),G contains a fractional[a,b]-factor covering e.A graph G is called a fractional(a,b,k)-critical covered graph if for any W■V(G)with|W|=k,G-W is fractional[a,b]-covered,which was first defined and investigated by Zhou,Xu and Sun[S.Zhou,Y.Xu,Z.Sun,Degree conditions for fractional(a,b,k)-critical covered graphs,Information Processing Letters 152(2019)105838].In this work,we proceed to study fractional(a,b,k)-critical covered graphs and derive a result on fractional(a,b,k)-critical covered graphs depending on minimum degree and neighborhoods of independent sets.

Unimodality of Independence Polynomials of the Cycle Cover Product of Graphs

An independent set in a graph G is a set of pairwise non-adjacent vertices.Let ik(G)denote the number of independent sets of cardinality k in G.Then its generating function I(G;x)=∑^(α(G))^(k=0)ik(G)x^(k),is called the independence polynomial of G(Gutman and Harary,1983).In this paper,we introduce a new graph operation called the cycle cover product and formulate its independence polynomial.We also give a criterion for formulating the independence polynomial of a graph.Based on the exact formulas,we prove some results on symmetry,unimodality,reality of zeros of independence polynomials of some graphs.As applications,we give some new constructions for graphs having symmetric and unimodal independence polynomials.