Degree conditions of induced matching extendable graphs

Abstract. A simple graph G is induced matching extendable,shortly IM-extendable,if every in-duced matching of G is included in a perfect matching of G. The degree conditions of IM-extend-able graphs are researched in this paper. The main results are as follows：     

导出匹配可扩图的局部运算

称图G为导出匹配图可扩的（简称为IM－可扩的），如果图G的一每个导出匹配都包含在G的一个完美匹配中，本给出了导出匹配可扩图的一些局部运算。     

路可扩图的一个充分条件

通过讨论图中任意一对不相邻顶点的度和,对路可扩图的充分条件进行研究,得到了如下结果：设图G的阶是n,如果G中任意一对不相邻顶点的度和至少为3/2n-1,则图G是路可扩的.并且说明了这里两不相邻顶点的度和的下界3/2n-1是最好可能的.     

Hamiltonian cycles in n‐extendable graphs

A graph G of order at least 2n+2 is said to be n‐extendable if G has a perfect matching and every set of n independent edges extends to a perfect matching in G. We prove that every pair of nonadjacent vertices x and y in a connected n‐extendable graph of order p satisfy deg_G x+deg_G y ≥ p ? n ? 1, then either G is hamiltonian or G is isomorphic to one of two exceptional graphs. © 2002 Wiley Periodicals, Inc. J Graph Theory 40: 75–82, 2002     

Bi-cycle extendable through a given set in balanced bipartite graphs

Let G=(X,Y;E) be a balanced bipartite graph of order 2n. The path-cover numberpc(H) of a graph H is the minimum number of vertex-disjoint paths that use up all the vertices of H. S⊆V(G) is called a balanced set of G if |S∩X|=|S∩Y|. In this paper, we will give some sufficient conditions for a balanced bipartite graph G satisfying that for every balanced set S, there is a bi-cycle of every length from |S|+2pc(〈S〉) up to 2n through S.     

3阶幂图的广义导出匹配可扩性

A graph G is induced matching extendable if every induced matching of G is included in a perfect matching of G. A graph G is generalized induced matching extendable if every induced matching of G is included in a maximum matching of G. A graph G is claw-free, if G dose not contain any induced subgraph isomorphic to K1,3. The k-th power of G, denoted by Gu, is the graph with vertex set V（G） in which two vertices are adjacent if and only if the distance between them is at most k in G. In this paper we show that, if the maximum matchings of G and G3 have the same cardinality, then G3 is generalized induced matching extendable. We also show that this result is best possible. As a result, we show that if G is a connected claw-flee graph, then G3 is generalized induced matching extendable.     

Characterization of the induced matching extendable graphs with 2n vertices and 3n edges

A graph $G$ is induced matching extendable or IM-extendable if every induced matching of $G$ is contained in a perfect matching of $G$. In 1998, Yuan proved that a connected IM-extendable graph on $2n$ vertices has at least $3n?2$ edges, and that the only IM-extendable graph with $2n$ vertices and $3n?2$ edges is $T\times K_2$ , where $T$ is an arbitrary tree on $n$ vertices. In 2005, Zhou and Yuan proved that the only IM-extendable graph with $2n\ge 6$ vertices and $3n?1$ edges is $T\times K_2+e$, where $T$ is an arbitrary tree on $n$ vertices and $e$ is an edge connecting two vertices that lie in different copies of $T$ and have distance 3 between them in $T\times K_2$. In this paper, we introduced the definition of $Q$-joint graph and characterized the connected IM-extendable graphs with $2n\ge 4$ vertices and $3n$ edges.     

Path extendable graphs

A pathP in a graphG is said to beextendable if there exists a pathP’ inG with the same endvertices asP such thatV(P)⊆V (P’) and |V(P’)|=|V(P)|+1. A graphG ispath extendable if every nonhamiltonian path inG is extendable. We investigate the extent to which known sufficient conditions for a graph to be hamiltonian-connected imply the extendability of paths in the graph. Several theorems are proved: for example, it is shown that ifG is a graph of orderp in which the degree sum of each pair of non-adjacent vertices is at leastp+1 andP is a nonextendable path of orderk inG thenk≤(p+1)/2 and 〈V (P)〉≅K _k orK _k −e. As corollaries of this we deduce that if δ(G)≥(p+2)/2 or if the degree sum of each pair of nonadjacent vertices inG is at least (3p−3)/2 thenG is path extendable, which strengthen results of Williamson [13].     

Extremal graphs having no matching cuts

A graph G = (V, E) is matching immune if there is no matching cut in G. We show that for any matching immune graph G, |E|≥?3(|V|?1)/2?. This bound is tight, as we define operations that construct, from a given vertex, exactly the class of matching immune graphs that attain the bound. © 2011 Wiley Periodicals, Inc. J Graph Theory 69:206‐222, 2012 .     

A note on minimal matching covered graphs

A graph is called matching covered if for its every edge there is a maximum matching containing it. It is shown that minimal matching covered graphs without isolated vertices contain a perfect matching.     

On the complexity of the dominating induced matching problem in hereditary classes of graphs

The dominating induced matching problem, also known as efficient edge domination, is the problem of determining whether a graph has an induced matching that dominates every edge of the graph. This problem is known to be NP-complete. We study the computational complexity of the problem in special graph classes. In the present paper, we identify a critical class for this problem (i.e., a class lying on a “boundary” separating difficult instances of the problem from polynomially solvable ones) and derive a number of polynomial-time results. In particular, we develop polynomial-time algorithms to solve the problem for claw-free graphs and convex graphs.     

Characterizations of maximum matching graphs of certain types

The maximum matching graph of a graph G is a graph whose vertices are maximum matchings of G and where two maximum matchings are adjacent in if they differ in exactly one edge. In this paper, the author characterizes the graphs whose maximum matching graphs are regular or cycles, and adds trees to the list of known maximum matching graphs.     

Maximum orders of extendable actions on surfaces

We determine the maximum order E_g of finite groups G acting on the closed surface Σ_g of genus g which extends over(S~3, Σ_g) for all possible embeddings Σ_g → S~3, where g 1.     

On perfect matchings in matching covered graphs

A graph is matching-covered if every edge of is contained in a perfect matching. A matching-covered graph is strongly coverable if, for any edge of , the subgraph is still matching-covered. An edge subset of a matching-covered graph is feasible if there exist two perfect matchings and such that , and an edge subset with at least two edges is an equivalent set if a perfect matching of contains either all edges in or none of them. A strongly matchable graph does not have an equivalent set, and any two independent edges of form a feasible set. In this paper, we show that for every integer , there exist infinitely many -regular graphs of class 1 with an arbitrarily large equivalent set that is not switching-equivalent to either or , which provides a negative answer to a problem of Lukot’ka and Rollová. For a matching-covered bipartite graph , we show that has an equivalent set if and only if it has a 2-edge-cut that separates into two balanced subgraphs, and is strongly coverable if and only if every edge-cut separating into two balanced subgraphs and satisfies and .     

树的导出匹配覆盖

The induced matching cover number of a graph G without isolated vertices,denoted by imc(G),is the minimum integer k such that G has k induced matchings M1,M2,…,Mk such that,M1∪M2 ∪…∪Mk covers V(G).This paper shows if G is a nontrivial tree,then imc(G) ∈ {△*0(G),△*0(G) + 1,△*0(G)+2},where △*0(G) = max{d0(u) + d0(v) :u,v ∈ V(G),uv ∈ E(G)}.     

Solution of a conjecture of Tewes and Volkmann regarding extendable cycles in in‐tournaments

A directed cycle C of a digraph D is extendable if there exists a directed cycle C′ in D that contains all vertices of C and an additional one. In 1989, Hendry defined a digraph D to be cycle extendable if it contains a directed cycle and every non‐Hamiltonian directed cycle of D is extendable. Furthermore, D is fully cycle extendable if it is cycle extendable and every vertex of D belongs to a directed cycle of length three. In 2001, Tewes and Volkmann extended these definitions in considering only directed cycles whose length exceed a certain bound 3≤k<n: a digraph D is k ‐extendable if every directed cycle of length t, where k≤t<n, is extendable. Moreover, D is called fully k ‐extendable if D is k ‐extendable and every vertex of D belongs to a directed cycle of length k. An in‐tournament is an oriented graph such that the in‐neighborhood of every vertex induces a tournament. This class of digraphs which generalizes the class of tournaments was introduced by Bang‐Jensen, Huang and Prisner in 1993. Tewes and Volkmann showed that every connected in‐tournament D of order n with minimum degree δ≥1 is ( ) ‐extendable. Furthermore, if D is a strongly connected in‐tournament of order n with minimum degree δ=2 or , then D is fully ( ) ‐extendable. In this article we shall see that if , every vertex of D belongs to a directed cycle of length , which means that D is fully ( ) ‐extendable. This confirms a conjecture of Tewes and Volkmann. © 2009 Wiley Periodicals, Inc. J Graph Theory 63: 82–92, 2010     

Forcing matching numbers of fullerene graphs

The forcing number or the degree of freedom of a perfect matching M of a graph G is the cardinality of the smallest subset of M that is contained in no other perfect matchings of G. In this paper we show that the forcing numbers of perfect matchings in a fullerene graph are not less than 3 by applying the 2-extendability and cyclic edge-connectivity 5 of fullerene graphs obtained recently, and Kotzig’s classical result about unique perfect matching as well. This lower bound can be achieved by infinitely many fullerene graphs.