Dingjun Lou

• PhD
• Professor (Full) at Sun Yat-sen University

A graph $ G $ with at least $ 2k $ vertices is called $ k $-subconnected if, for any $ 2k $ vertices in $ G $, there are $ k $ independent paths $ P_{1}, P_{2}, \cdots, P_{k} $ joining the $ 2k $ vertices in pairs. A graph $ G $ is minimally 2-subconnected if $ G $ is $ 2 $-subconnected and $ G-e $ is not $ 2 $-subconnected for any edge e in G. The...

A graph G with at least 2k vertices is called k-subconnected if, for any 2k vertices x1,x2,⋯,x2k in G, there are k independent paths joining the 2k vertices in pairs in G. In this paper, we prove that a k-connected planar graph with at least 2k vertices is k-subconnected for k=1,2; a 4-connected planar graph is k-subconnected for each k such that 1...

Cyclic (vertex and edge) connectivity is an important concept in graphs. While cyclic edge connectivity (cλ) has been studied for many years, the study at cyclic vertex connectivity (cκ) is still at the initial stage. And cκ seems to be more complicated than cλ. We have got a sufficient condition that ν(G)≥2g(k−1) for cκ≠∞. On the other hand, if ν(...

Let G be a graph, \(\nu \) the order of G and k a positive integer such that \(k\le (\nu -2)/2\). Then G is said to be k-extendable if it has a matching of size k and every matching of size k extends to a perfect matching of G. A graph G is Hamiltonian if it contains a Hamiltonian cycle. A graph G is Hamiltonian-connected if, for any two of its ver...

A digraph D with n vertices is Hamiltonian (pancyclic and vertex‐pancyclic, respectively) if D contains a Hamilton cycle (a cycle of every length 3,4,…,n, for every vertex v∈V(D), a cycle of every length 3,4,…,n through v, respectively.) It is well‐known that a strongly connected tournament is Hamiltonian, pancyclic, and vertex pancyclic. A digraph...

In this paper, we give the definition of k-subconnected graphs. Let G be a graph with at least 3k − 1 vertices. Then G is called a k-subconnected graph if, for any 2k vertices of G, there are k disjoint paths in G joining them in pairs. We prove that G is a k-subconnected graph if and only if, for any cutset S ⊆ V(G) with |S| ≤ k − 1, ω(G − S) ≤ |S...

For a connected graph G, a set S of vertices is a cyclic vertex cutset if G-S\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$G - S$$\end{document} is not connected and...

For a connected graph G, a set S of vertices is a cyclic vertex cutset if G-S\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$G - S$$\end{document} is not connected and...

We study $M$-alternating Hamilton paths and $M$-alternating Hamilton cycles in a simple connected graph $G$ on $\nu$ vertices with a perfect matching $M$. Let $G$ be a bipartite graph, we prove that if for any two vertices $x$ and $y$ in different parts of $G$, $d(x)+d(y)\geq \nu/2+2$, then $G$ has an $M$-alternating Hamilton cycle. For general gra...

We determine the minimum size of $n$-factor-critical graphs and that of $k$-extendable bipartite graphs, by considering Harary graphs and related graphs. Moreover, we determine the minimum size of $k$-extendable non-bipartite graphs for $k=1,\ 2$, and pose a related conjecture for general $k$.

In the study of cycles and paths, the meta-conjecture of Bondy that sufficient conditions for Hamiltonicity often imply pancyclicity has motivated research on the existence of cycles and paths of many lengths. Hendry further introduced the stronger concepts of cycle extendability and path extendability, which require that every cycle or path can be...

For a connected graph G, a set S of vertices is a cyclic vertex cutset if G−S is not connected and at least two components contain a cycle respectively. The cyclic vertex connectivity cκ(G) is the cardinality of a minimum cyclic vertex cutset. In this paper, for any k-regular graph G with girth g and the number v of vertices, we give a sufficient c...

Let B(G) denote the bipartite double cover of a non-bipartite graph G with v ⩾ 2 vertices and ɛ 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 1-extendable graph if and only if G is a minimally perfect 2-matching covered graph and for each e =...

In this paper, we develop an 0(k9V6) time algorithm to determine the cyclic edge connectivity of κ-regular graphs of order V for κ ≥ 3 which is an improvement of a known algorithm by Lou and Wang.

We determine the minimum size of n-factor-critical graphs and that of k-extendable bipartite graphs, by considering Harary graphs and related graphs. Moreover, we determine the minimum size of k-extendable non-bipartite graphs for k = 1, 2, and pose a related conjecture for general k.

A near perfect matching is a matching covering all but one vertex in a graph. Let G be a connected graph and n≤(|V(G)|−2)/2 be a positive integer. If any n independent edges in G are contained in a near perfect matching, then G is said to be defectn-extendable. In this paper, we first characterize defect n-extendable bipartite graph G with n=1 or κ...

In this paper, we obtain necessary and sufficient conditions for a graph G not to have an M-alternating path between two vertices in G.

In this paper, we introduce a corresponding between bipartite graphs with a perfect matching and digraphs, which implicates an equivalent relation between the extendibility of bipartite graphs and the strongly connectivity of digraphs. Such an equivalent relation explains the similar results on $k$-extendable bipartite graphs and $k$-strong digraph...

In this paper, we give a sufficient and necessary condition for a $k$-extendable graph to be $2k$-factor-critical when $k=\nu/4$, and prove some results on independence numbers in $n$-factor-critical graphs and $k\frac{1}{2}$-extendable graphs. Comment: This paper has been published on Ars Combinatoria

Let G be a graph with vertex set V(G). Let n, k, d be non-negative integers such that n+2k+d≤|V(G)|−2 and |V(G)|−n−d are even. A matching which saturates exactly |V(G)|−d vertices is called a defect-d matching of G. If when deleting any n vertices the remaining subgraph contains a matching of k edges and every k-matching can be extended to a defect...

We study M-alternating Hamilton paths, and M-alternating Hamilton cycles in a simple connected graph G on ν vertices with a perfect matching M. Let G be a bipartite graph, we prove that if for any two vertices x and y in different parts of G, d(x)+d(y)≥ν/2+2, then G has an M-alternating Hamilton cycle. For general graphs, a condition for the existe...

A near perfect matching is a matching saturating all but one vertex in a graph. Let G be a connected graph. If any n independent edges in G are contained in a near perfect matching where n is a positive integer and n⩽(|V(G)|-2)/2, then G is said to be defect n-extendable. If deleting any k vertices in G where k⩽|V(G)|-2, the remaining graph has a p...

In this paper, it is proved that let G be a bipartite graph with bipartition (X,Y) and with a perfect matching M, let G be an n-extendable graph, then G is minimally n-extendable if and only if, for any two vertices x∈X and y∈Y such that xy∈E(G), there are exactly n internally disjoint (x,y)M-alternating paths P1,P2,…,Pn such that Pi(1⩽i⩽n) starts...

In this paper, we characterize the graphs with infinite cyclic edge connectivity. Then we design an efficient algorithm to determine whether a graph has finite cyclic edge connectivity or infinite cyclic edge connectivity.

This paper deals with the problem of construction hamiltonian paths of optimal weights in Halin graphs. There are three versions of the hamiltonian path: none or one or two of endvertices are specified. We present O(|V|) algorithms to all the versions of the problem.

In this paper, we give a sufficient and necessary condition for a $k$-extendable graph to be $2k$-factor-critical when $k=\nu/4$, and prove some results on independence numbers in $n$-factor-critical graphs and $k\frac{1}{2}$-extendable graphs.

In this paper, we show that if $k\geq (\nu+2)/4$, where $\nu$ denotes the order of a graph, a non-bipartite graph $G$ is $k$-extendable if and only if it is $2k$-factor-critical. If $k\geq (\nu-3)/4$, a graph $G$ is $k\ 1/2$-extendable if and only if it is $(2k+1)$-factor-critical. We also give examples to show that the two bounds are best possible...

A near perfect matching is a matching saturating all but one vertex in a graph. If G is a connected graph and any n independent edges in G are contained in a near perfect matching where n is a positive integer and n⩽(|V(G)|-2)/2, then G is said to be defect n-extendable. This paper first shows that the connectivity of defect n-extendable bipartite...

In this paper, we consider whether the vertices of graph G can be partitioned into K subsets V 1, V 2, ... V K so that for each i ∈ {1,...,K}, the subgraph induced by V i is a perfect matching where K ≤ ∣ V ∣. It is known that it is an NP complete problem in general graphs. In this paper, we restrict the problem in Halin graphs and give an algorith...

This paper studies the problem of making a bipartite graph 1-extendable by adding the smallest number of new edges that preserve bipar- titeness. Let G = (V,E) be a graph with at least 2k+ 2 vertices. A matching is a subset of E(G) of which the edges are disjoint. A matching with k edges is called to be A;-matching. If G has a fc-matching and every...

Finding a matching with the maximum total weight is the wellkonwn assignment problem of assigning people to jobs and maximize the profits. In this paper, we focus on finding the maximum-weighted matching in Halin graphs. First, we review the method of "shrinking fan". Second, we show how to use the method in finding the maximum-weighted matching in...

In this paper, we show that if $k\geq (\nu+2)/4$, where $\nu$ denotes the order of a graph, a non-bipartite graph $G$ is $k$-extendable if and only if it is $2k$-factor-critical. If $k\geq (\nu-3)/4$, a graph $G$ is $k\ 1/2$-extendable if and only if it is $(2k+1)$-factor-critical. We also give examples to show that the two bounds are best possible...

We develop a polynomial time algorithm to determine the cyclic edge connectivity of a k-regular graph for k≥3. Cyclic edge connectivity is defined as the size of a minimum edge cutset such that at least two components of the separated graph contain a cycle.

Let G be a balanced bipartite graph with partite sets X and Y, which has a perfect matching, and let x∈X and y∈Y. Let k be a positive integer. Then we prove that if G has k internally disjoint alternating paths between x and y with respect to some perfect matching, then G has k internally disjoint alternating paths between x and y with respect to e...

Let G be a graph with even order. Let M be a matching in G and x1,x2,…,x2r, be the M-unsaturated vertices in G. Then G has a perfect matching if and only if there are r independent M-augmenting paths joining the 2r vertices in pairs. Let G be a graph with a perfect matching M. It is proved that G is 2k-critical if and only if for any 2k vertices u1...

A minimum degree condition is given for a bipartite graph to contain a 2-factor each component of which contains a previously specified vertex. © 2004 Wiley Periodicals, Inc. J Graph Theory 46: 145–166, 2004

Let G be a graph with a perfect matching M. In this paper, we prove two theorems to characterize the graph G in which there is no M-alternating path between two vertices x and y in G.

Let G be a simple connected graph on 2n vertices with perfect matching. For a given positive integer k (0 6 k 6 n − 1), G is k-extendable if any matching of size k in G is contained in a perfect matching of G. It is proved that if G is a k-extendable graph on 2n vertices with k ¿ n=2, theneither G is bipartite or the connectivity of G is at least 2...

Let n be a non-negative integer. A graph G is said to be n-matchable if the subgraph G S has a perfect matching for any subset S of V (G) with |S| = n. In this paper, we obtain sucient conditions for dierent classes of graphs to be n- matchable. Since 2k-matchable graphs must be k-extendable, we have generalized the results about k-extendable graph...

In this paper, we show a necessary and sufficient condition which char-acterizes all factor critical graphs. Using this necessary and sufficient condition, we develop a linear time algorithm to determine whether a graph is factor critical if one of its maximum matchings is given.

Let G be a graph with a perfect matching M 0. It is proved that G is 1-extendable if and only if for any pair of vertices x and y with an M 0-alternating xy-path P 0 of length three which starts with an edge that belongs to M 0, there exists an M 0-alternating path P connecting x and y, of which the starting and the ending edges do not belong to M...

Let G be a bipartite graph with bipartition (X,Y) which has a perfect matching. It is proved that G is n-extendable if and only if for any perfect matching M of G and for each pair of vertices x in X and y in Y there are n internally disjoint M-alternating paths connecting x and y. Furthermore, these n paths start and end with edges in E(G)⧹M. This...

In this paper, we show a necessary and sufficient condition which characterizes all bicritical graphs. Using this necessary and sufficient condition we develop a highly efficient algorithm to determine whether a graph is bicritical, of which the time complexity is bounded by O (|V| |E|), and this is the best result that has ever been known.

In this paper, we develop a polynomial time algorithm to find out all the minilnum cyclic edge cutsets of a 3-regular graph, and therefore to determine the cyclic edge connectivity of a cubic graph. The algorithm is recursive, with complexity bounded by O(n3log2 n). The algorithm shows that the number of mini~um cyclic edge cut sets of a 3-regular...

It is proved that, in a minimal n-extendable bipartite graph, the subgraph induced by the edges both ends of which have degree at least n + 2 is a forest. As a consequence, every minimal n-extendable bipartite graph has at least 2n + 2 vertices of degree n + 1. This result is sharp. Some other structural results on minimally n-extendable bipartite...

Let G be a graph and v ∈ V(G). Let Nk(v) = {u | u ∈ V(G) and d(u, v) = k}. It is proved that if G is a connected graph with ∞ > g(G) ⩾ 4 and with even order and if, for each vertex v in V(G), α(G[N2(v)]) ⩽ d(v) − 1, then G is regular and ⌈d(v)/4⌉-extendable. All results in this paper are sharp.

Let G be a connected graph with ν ≥ 3. Let v ∈ V(G). We define Nk(v) = {u|u ∈ V(G) and d(u,v) = k}. It is proved that if for each vertex v ∈ V(G) and for each independent set S ⊆ N2(v), |N(S) ∩ N(v)| ≥ |S| + 1, then G is hamiltonian. Several previously known sufficient conditions for hamiltonian graphs follow as corollaries. It is also proved that...

We prove the following: Let G be connected balanced bipartite graph of order 2n≥4. If G satisfies the localization condition |N 2 (u)∖N(v)|+2≤d(u) for any u,v∈V(G) with d(u,v)=3, where N 2 (u)={w∈V(G)∣d(u,w)=2}, then G is either bipancyclic or isomorphic to C 6 . Furthermore, a conjecture is proposed.

It is proved that if G is a triangle-free graph with v vertices whose independence number does not exceed its connectivity then G has cycles of every length n for 4 ⩽ n ⩽ v(G) unless G = Kv/2,v/2 or G is a 5-cycle. This was conjectured by Amar, Fournier and Germa.

It is proved that every connected strongly regular graph (v(G),k,α,β) with even order is 2-extendable when k ⩾ 3, except the Petersen graph, the (6, 4, 2, 4) graph and K4.

It is proved that a cyclically (k − 1)(2n − 1)-edge-connected edge transitive k-regular graph with even order is n-extendable, where k ≥ 3 and k − 1 ≥ n ≥ ⌈(k + 1)/2⌉. The bound of cyclic edge connectivity is sharp when k = 3. © 1993 John Wiley & Sons, Inc.

A cyclically m-edge-connected n-connected k-regular graph is called an (m.n.k) graph. It is proved that for any m > 0 and k ⩾3, there is an (m, k, k) bipartite graph. A graph G is n-extendable if every matching of size n in G lies in a perfect matching of G. We prove the existence of a (k2 −1, k + 1, k + 1) bipartite graph which is not k-extendable...

Some sufficient conditions for the 2-extendability of k-connected k-regular (k⩾3) planar graphs are given. In particular, it is proved that for k⩾3, a k-connected k-regular planar graph with each cyclic cutset of sufficiently large size is 2-extendable.

It is a well-known fact that the linear arboricity of a k-regular graph is for k = 3,4. In this paper, we prove that if the number of edges of a k-regular graph is divisible by , then its edge set can be partitioned into linear forests, all of which have the same number of edges (k = 3,4).

A graph G that has a perfect matching is n-extendable if every matching of size n lies in a perfect matching of G. We show that when the connectivity of a line graph, power graph, or total graph is sufficiently large then it is n-extendable. Specifically: if G has even size and is (2n + 1)-edge-connected or (n + 2)-connected, then its line graph is...

We study the problem of finding the (n,k,0)-extendability in bipartite graphs. Let G be a graph with vertex set V(G). Let n,k,d be non-negative integers such that n + 2k + d ≤ |V(G)| − 2 and |V(G)| − n − d is even. A matching which saturates exactly |V(G)| − d vertices of G is called a defect-d matching of G. If when deleting any n vertices in V(G)...

A near perfect matching is a matching saturating all but one vertex in a graph. If every edge in G is contained in a near perfect matching of G, then G is defect 1-extendable. Let G=(U,W) be a bipartite graph with |W|=|U|+1 and |U|≥2 where U∪W is a bipartition of G. First, it is proved that G is defect 1-extendable if and only if G ' is 1-extendabl...