Bipartite graphs with a perfect matching and digraphs

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 digraphs. We also study the relation among $k$-extendable bipartite graphs, $k$-strong digraphs and combinatorial matrices. For bipartite graphs that are not 1-extendable and digraphs that are not strong, we prove that the elementary components and strong components are counterparts.Comment: 2 figures, 7 pages. A short version of this paper is published in "Advances and applications in Discrete Mathematics". This version contains some interesting comparison between $k$-strong digraphs and $k$-extendable bigraphs, which is deleted in the published versio

Matching Connectivity: On the Structure of Graphs with Perfect Matchings

We introduce the concept of matching connectivity as a notion of connectivity in graph admitting perfect matchings which heavily relies on the structural properties of those matchings. We generalise a result of Robertson, Seymour and Thomas for bipartite graphs with perfect matchings (see [Neil Roberts, Paul D Seymour, and Robin Thomas. Permanents, pfaffian orientations, and even directed curcuits. Annals of Mathematics, 150(2):929-975, 1999]) in order to obtain a concept of alternating paths that turns out to be sufficient for the description of our connectivity parameter. We introduce some basic properties of matching connectivity and prove a Menger-type result for matching n-connected graphs. Furthermore, we show that matching connectivity fills a gap in the investigation of n-extendable graphs and their connectivity properties. To be more precise we show that every n-extendable graph is matching n-connected and for the converse every matching (n+1)-connected graph either is n-extendable, or belongs to a well described class of graphs: the brace h-critical graphs.Comment: Error in main proo

Minimum size of n-factor-critical graphs and k-extendable graphs

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 $k$.Comment: 13 pages, 8 figures, published in Graphs and Combinatoric

Equivalence between Extendibility and Factor-Criticality

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. Our results are answers to a problem posted by Favaron [3] and Yu [11].Comment: This paper has been published at Ars Combinatori

Surface embedding of non-bipartite kk-extendable graphs

We find the minimum number $k=\mu'(\Sigma)$ for any surface $\Sigma$, such that every $\Sigma$-embeddable non-bipartite graph is not $k$-extendable. In particular, we construct the so-called bow-tie graphs $C_6\bowtie P_n$, and show that they are $3$-extendable. This confirms the existence of an infinite number of $3$-extendable non-bipartite graphs which can be embedded in the Klein bottle.Comment: 17 pages, 5 figure

Equistarable bipartite graphs

Recently, Milani\v{c} and Trotignon introduced the class of equistarable graphs as graphs without isolated vertices admitting positive weights on the edges such that a subset of edges is of total weight $1$ if and only if it forms a maximal star. Based on equistarable graphs, counterexamples to three conjectures on equistable graphs were constructed, in particular to Orlin's conjecture, which states that every equistable graph is a general partition graph. In this paper we characterize equistarable bipartite graphs. We show that a bipartite graph is equistarable if and only if every $2$-matching of the graph extends to a matching covering all vertices of degree at least $2$. As a consequence of this result, we obtain that Orlin's conjecture holds within the class of complements of line graphs of bipartite graphs. We also connect equistarable graphs to the triangle condition, a combinatorial condition known to be necessary (but in general not sufficient) for equistability. We show that the triangle condition implies general partitionability for complements of line graphs of forests, and construct an infinite family of triangle non-equistable graphs within the class of complements of line graphs of bipartite graphs

Minimum k-critical bipartite graphs

We study the problem of Minimum $k$-Critical Bipartite Graph of order $(n,m)$ - M$k$CBG-$(n,m)$: to find a bipartite $G=(U,V;E)$, with $|U|=n$, $|V|=m$, and $n>m>1$, which is $k$-critical bipartite, and the tuple $(|E|, \Delta_U, \Delta_V)$, where $\Delta_U$ and $\Delta_V$ denote the maximum degree in $U$ and $V$, respectively, is lexicographically minimum over all such graphs. $G$ is $k$-critical bipartite if deleting at most $k=n-m$ vertices from $U$ creates $G'$ that has a complete matching, i.e., a matching of size $m$. We show that, if $m(n-m+1)/n$ is an integer, then a solution of the M$k$CBG-$(n,m)$ problem can be found among $(a,b)$-regular bipartite graphs of order $(n,m)$, with $a=m(n-m+1)/n$, and $b=n-m+1$. If $a=m-1$, then all $(a,b)$-regular bipartite graphs of order $(n,m)$ are $k$-critical bipartite. For $a<m-1$, it is not the case. We characterize the values of $n$, $m$, $a$, and $b$ that admit an $(a,b)$-regular bipartite graph of order $(n,m)$, with $b=n-m+1$, and give a simple construction that creates such a $k$-critical bipartite graph whenever possible. Our techniques are based on Hall's marriage theorem, elementary number theory, linear Diophantine equations, properties of integer functions and congruences, and equations involving them

On Perfect Matchings in Matching Covered Graphs

Let $G$ be a matching-covered graph, i.e., every edge is contained in a perfect matching. An edge subset $X$ of $G$ is feasible if there exists two perfect matchings $M_1$ and $M_2$ such that $|M_1\cap X|\not\equiv |M_2\cap X| \pmod 2$. Lukot'ka and Rollov\'a proved that an edge subset $X$ of a regular bipartite graph is not feasible if and only if $X$ is switching-equivalent to $\emptyset$, and they further ask whether a non-feasible set of a regular graph of class 1 is always switching-equivalent to either $\emptyset$ or $E(G)$? Two edges of $G$ are equivalent to each other if a perfect matching $M$ of $G$ either contains both of them or contains none of them. An equivalent class of $G$ is an edge subset $K$ with at least two edges such that the edges of $K$ are mutually equivalent. An equivalent class is not a feasible set. Lov\'asz proved that an equivalent class of a brick has size 2. In this paper, we show that, for every integer $k\ge 3$, there exist infinitely many $k$-regular graphs of class 1 with an arbitrarily large equivalent class $K$ such that $K$ is not switching-equivalent to either $\emptyset$ or $E(G)$, which provides a negative answer to the problem proposed by Lukot'ka and Rollov\'a. Further, we characterize bipartite graphs with equivalent class, and characterize matching-covered bipartite graphs of which every edge is removable.Comment: 10 pages, 3 figure

On the restricted matching of graphs in surfaces

A connected graph $G$ with at least $2m+2n+2$ vertices is said to have property $E(m,n)$ if, for any two disjoint matchings $M$ and $N$ of size $m$ and $n$ respectively, $G$ has a perfect matching $F$ such that $M\subseteq F$ and $N\cap F=\varnothing$. In particular, a graph with $E(m,0)$ is $m$-extendable. Let $\mu(\Sigma)$ be the smallest integer $k$ such that no graphs embedded on a surface $\Sigma$ are $k$-extendable. Aldred and Plummer have proved that no graphs embedded on the surfaces $\Sigma$ such as the sphere, the projective plane, the torus, and the Klein bottle are $E(\mu(\Sigma)-1,1)$. In this paper, we show that this result always holds for any surface. Furthermore, we obtain that if a graph $G$ embedded on a surface has sufficiently many vertices, then $G$ has no property $E(k-1,1)$ for each integer $k\geq 4$, which implies that $G$ is not $k$-extendable. In the case of $k=4$, we get immediately a main result that Aldred et al. recently obtained.Comment: 9 page

Notes on factor-criticality, extendibility and independence number

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 Combinatori