Related papers: Tree Matchings

Luo, Tian and Wu (2022) conjectured that for any tree $T$ with bipartition $X$ and $Y$, every $k$-connected bipartite graph $G$ with minimum degree at least $k+t$, where $t=$max$\{|X|,|Y|\}$, contains a tree $T'\cong T$ such that $G-V(T')$…

Let $f_k(n,H)$ denote the maximum number of edges not contained in any monochromatic copy of~$H$ in a $k$-coloring of the edges of $K_n$, and let $ex(n,H)$ denote the Tur\'an number of $H$. In place of $f_2(n,H)$ we simply write $f(n,H)$.…

Given an integer weighted bipartite graph $\{G=(U\sqcup V, E), w:E\rightarrow \mathbb{Z}\}$ we consider the problems of finding all the edges that occur in some minimum weight matching of maximum cardinality and enumerating all the minimum…

Motivated by the concept of well-covered graphs, we define a graph to be well-bicovered if every vertex-maximal bipartite subgraph has the same order (which we call the bipartite number). We first give examples of them, compare them with…

Let $G$ be a nontrivial connected graph of order $n$ with an edge-coloring $c:E(G)\rightarrow\{1,2,\dots,t\}$,$t\in\mathbb{N}$, where adjacent edges may be colored with the same color. A tree $T$ in $G$ is a \emph{proper tree} if no two…

We consider a matching problem in a bipartite graph $G$ where every vertex has a capacity and a strict preference order on its neighbors. Furthermore, there is a cost function on the edge set. We assume $G$ admits a perfect matching, i.e.,…

A strong edge-coloring $\varphi$ of a graph $G$ assigns colors to edges of $G$ such that $\varphi(e_1)\ne \varphi(e_2)$ whenever $e_1$ and $e_2$ are at distance no more than 1. It is equivalent to a proper vertex coloring of the square of…

In this paper and a companion paper, we prove that, if $m$ is sufficiently large, every graph on $m+1$ vertices that has a universal vertex and minimum degree at least $\lfloor \frac{2m}{3} \rfloor$ contains each tree $T$ with $m$ edges as…

It is well known that a graph $G$ has a symmetric spectrum if and only if it is bipartite, a signed graph $\Gamma=(G,\sigma)$ has a symmetric spectrum if $G$ is bipartite. However, there exists a spectrally symmetric signed graph…

Let $G$ be a graph and $k$ be a positive integer, and let $Kc(G, k)$ denote the number of Kempe equivalence classes for the $k$-colorings of $G$. In 2006, Mohar noted that $Kc(G, k) = 1$ if $G$ is bipartite. As a generalization, we show…

In a graph $G$, the $2$-neighborhood of a vertex set $X$ consists of all vertices of $G$ having at least $2$ neighbors in $X$. We say that a bipartite graph $G(A,B)$ satisfies the double Hall property if $|A|\geq2$, and every subset $X…

Wu, Zhang and Li [4] conjectured that the set of vertices of any simple graph $G$ can be equitably partitioned into $\lceil(\Delta(G)+1)/2\rceil$ subsets so that each of them induces a forest of $G$. In this note, we prove this conjecture…

The bipartite Tur\'{a}n number of a graph $H$, denoted by $ex(m,n; H)$, is the maximum number of edges in any bipartite graph $G=(X,Y; E)$ with $|X|=m$ and $|Y|=n$ which does not contain $H$ as a subgraph. In this paper, we determined…

The Tree Decomposition Conjecture by Bar\'at and Thomassen states that for every tree $T$ there exists a natural number $k(T)$ such that the following holds: If $G$ is a $k(T)$-edge-connected simple graph with size divisible by the size of…

We show that the number of $k$-matching in a given undirected graph $G$ is equal to the number of perfect matching of the corresponding graph $G_k$ on an even number of vertices divided by a suitable factor. If $G$ is bipartite then one can…

An edge $e$ of a matching covered graph $G$ is removable if $G-e$ is also matching covered. The notion of removable edge arises in connection with ear decompositions of matching covered graphs introduced by Lov\'asz and Plummer. A…

A bipartite graph is called bipancyclic if it contains cycles of every even length from four up to the number of vertices in the graph. A theorem of Schmeichel and Mitchem states that for $n \geq 4$, every balanced bipartite graph on $2n$…

Let $k$ be a positive integer. Let $G$ be a balanced bipartite graph of order $2n$ with bipartition $(X, Y)$, and $S$ a subset of $X$. Suppose that every pair of nonadjacent vertices $(x,y)$ with $x\in S, y\in Y$ satisfies $d(x)+d(y)\geq…

For a given graph, the unlabeled subgraphs $G-v$ are called the cards of $G$ and the deck of $G$ is the multiset $\{G-v: v \in V(G)\}$. Wendy Myrvold [Ars Combinatoria, 1989] showed that a non-connected graph and a connected graph both on…

We prove that for every $k$ and every $\varepsilon>0$, there exists $g$ such that every graph with tree-width at most $k$ and odd-girth at least $g$ has circular chromatic number at most $2+\varepsilon$.