Mao-cheng Cai

Connected [k, k + 1]-factors of graphs

Abstract Let k be an odd integer ⩾ 3, and G be a connected graph of odd order n with n ⩾ 4k − 3, and minimum degree at least k. In this paper it is proved that if for each pair of nonadjacent vertices u, v in G max {d G (u), d G (v)}⩾n/2, then G has an almost k±-factor F± and a matching M such that F− and M are edge-disjoint and F− + M is a connected [k, k + 1]-factor of G (an almost k±-factor F± is a factor that every vertex has degree k except at most one with degree k ± 1). As an immediate consequence, the result gives a solution to a problem of Kano on the existence of connected [k, k + 1]-factors

A remark on the number of vertices of degree k in a minimally k -edge-connected graph

Abstract Let G be a minimally k-edge-connected simple graph and vk(G) be the number of vertices of degree k in G. Mader (1974) proved that (i) v k (G) ⩾ ⌊ (|G|−1) (2k + 1) ⌋ + k + 1 for even k, and (ii) v k (G)⩾⌊ |G| (k + 1) ⌋ + k for odd k⩾5 and v k (G)⩾⌊ 2|G| (k + 1) ⌋ + k − 2 for odd k⩾7, where |G| denotes the number of vertices of G. In this paper we slightly improve the result for k being even, i.e., v k (G)⩾⌊ |G| (k + 1) ⌋ + k if k⩾4 and v k (G)⩾⌊ 2|G| (k + 1) ⌋ + k−2 if k⩾10.

Solution to a problem on degree sequences of graphs

Abstract Let a 1 ,a 2 ,…,a n and b 1 ,b 2 ,…,b n be integers with 0⩽a i ⩽b i for i=1,2,…,n . The purpose of this note is to give a good characterization for the existence of a simple graph G with vertices v 1 ,v 2 ,…,v n such that a i ⩽d G (v i )⩽b i for i=1,2,…,n . This solves a research problem posed by Niessen and generalizes an Erdős–Gallai theorem.

Upper minus domination in regular graphs

Abstract A three-valued function f defined on the vertices of a graph G=(V,E),f : V→{−1,0,1} , is a minus dominating function if the sum of its function values over any closed neighborhood is at least one. That is, for every v∈V, f(N[v])⩾1 , where N[v] consists of v and every vertex adjacent to v . The weight of a minus dominating function is f(V)=∑ v∈V f(v) . The upper minus domination number of a graph G , denoted Γ − (G) , equals the maximum weight of a minimal minus dominating function of G . In this paper, sharp upper bounds on Γ − of regular graphs are found. Thus, we answer an open problem proposed by Henning and Slater (Discrete Math. 158 (1996) 87–98).

Upper bounds for the K-subdomination number of graphs

For a positive integer k, a k-subdominating function of G=(V,E) is a function f:V → {-1, 1} such that the sum of the function values, taken over closed neighborhoods of vertices, is at least one for at least k vertices of G. The sum of the function values taken over all vertices is called the aggregate of f and the minimum aggregate among all k-subdominating functions of G is the k-subdomination number γks(G). In this paper, we solve a conjecture proposed in (Ars. Combin 43 (1996) 235), which determines a sharp upper bound on γks(G) for trees if k > |V|/2 and give an upper bound on γks(G) for connected graphs.

The maximal size of graphs with at most k edge-disjoint paths connecting any two adjacent vertices

Let n and k be positive integers satisfying k+1?n?3k ? 1, and G a simple graph of order n and size e(G) with at most k edge-disjoint paths connecting any two adjacent vertices. In this paper we prove that e(G)??(n+k)2/8?, and give complete characterizations of the extremal graphs and the extremal minimally k-edge-connected graphs.

On quasifactorability in graphs

Abstract Given a graph G and two functions f and g:V(G)→ Z + with f(v)⩾g(v) for each v∈V(G) , a (g,f) -quasifactor in G is a subgraph Q of G such that for each vertex v in V(Q), g(v)⩽d Q (v)⩽f(v) ; in the particular case when ∀v∈V(Q), f(v)=g(v)=k∈ N , we say that Q is a k -quasifactor. A subset S of vertices of G is said (g,f) -quasifactorable in G if there exists some (g,f) -quasifactor that contains all the vertices of S . In this paper, we give several results on the 2 -quasifactorability of a vertex subset which are related to minimum degree, degree sum, independence number and neighborhood union conditions.

Vertices of degree k in a minimally k -edge-connected digraph

Abstract Let k be a positive integer and D=(V,E) be a minimally k -edge-connected simple digraph. For a vertex x∈V(D) , its outdegree δ + (x) (indegree δ − (x) ) is the number of edges leaving (entering) x . Let u + (D) (resp. u ± (D) and u − (D) ) denote the number of vertices x in D such that δ + (x)=k − (x) (resp. δ + (x)=δ − (x)=k and δ + (x)>k=δ − (x) ). In this paper we prove that u + (D)+2u ± (D)+u − (D)⩾2k+2, which was conjectured by Mader (Combinatorics 2 (1996) 423–449). We also present a lower bound on u + (D)+u ± (D)+u − (D) when |D|⩾4k−1 .

An algorithm for optimum common root functions of two digraphs

Abstract Let G1 and G2 be finite digraphs, both with vertex set V. Suppose that each v of V has nonnegative integers f(v) and g(v) with f(v)⩽g(v), and each arc e of Gi has nonnegative integers ai(e) and bi(e) with ai(e)⩽bi(e), i=1,2. In Cai (1990) a necessary and sufficient condition was given for the existence of k arborescences in Gi covering each arc e of Gi at least ai(e) and at most bi(e) times, i=1,2, and satisfying the condition that for each v in V f(v)⩽r 1 (v)=r 2 (v)⩽g(v) , where ri(v) denotes the number of the arborescences in Gi rooted at v. Such an ri is called a common root function and denoted by r. In this paper, we present a polynomial algorithm for finding an optimum common root function r for a given weight function defined on V.