Related papers: A precise condition for independent transversals i…

A bi-hole of size $t$ in a bipartite graph $G$ is a copy of $K_{t,t}$ in the bipartite complement of $G$. Given an $n \times n$ bipartite graph $G$, let $\beta(G)$ be the largest $k$ for which $G$ has a bi-hole of size $k$. We prove that \[…

A {\em brick} is a non-bipartite matching covered graph without non-trivial tight cuts. Bricks are building blocks of matching covered graphs. We say that an edge $e$ in a brick $G$ is {\em $b$-invariant} if $G-e$ is matching covered and a…

Let $i_t(G)$ be the number of independent sets of size $t$ in a graph $G$. Alavi, Erd\H{o}s, Malde and Schwenk made the conjecture that if $G$ is a tree then the independent set sequence $\{i_t(G)\}_{t\geq 0}$ of $G$ is unimodal; Levit and…

Given a partition $V_1 \sqcup V_2 \sqcup \dots \sqcup V_m$ of the vertex set of a graph, we are interested in finding multiple disjoint independent sets that contain the correct fraction of vertices of each $V_j$. We give conditions for the…

A graph is said to be edge-transitive if its automorphism group acts transitively on its edges. It is known that edge-transitive graphs are either vertex-transitive or bipartite. In this paper we present a complete classification of all…

A graph $G = (V, E)$ is said to be word-representable if there exists a word $w$ over the alphabet $V$ such that, for any two distinct letters $x, y \in V$, the letters $x$ and $y$ alternate in $w$ if and only if $xy \in E$. A graph is…

A vertex partition in which every part induces a 2-connected subgraph is called a 2-proper partition. This concept was introduced by Ferrara et al. in 2013, and Borozan et al. gave the best possible minimum degree condition for the…

A bipartite graph $G = (X \cup Y, E)$ is a 2-layer $k$-planar graph if it admits a drawing on the plane such that the vertices in $X$ and $Y$ are placed on two parallel lines respectively, edges are drawn as straight-line segments, and…

Given a collection $\mathcal{G}=(G_1,\dots, G_h)$ of graphs on the same vertex set $V$ of size $n$, an $h$-edge graph $H$ on the vertex set $V$ is a $\mathcal{G}$-transversal if there exists a bijection $\lambda : E(H) \rightarrow [h]$ such…

R. Wang (Discrete Mathematics and Theoretical Computer Science, vol. 19(3), 2017) proposed the following problem. \textbf{Problem.} Let $D$ be a strongly connected balanced bipartite directed graph of order $2a\geq 8$. Suppose that…

A digraph $D$ with a subset $S$ of $V(D)$ is called $\boldsymbol{S}${\bf -strong} if for every pair of distinct vertices $u$ and $v$ of $S$, there is a $(u, v)$-dipath and a $(v, u)$-dipath in $D$. We define a digraph $D$ with a subset $S$…

A famous conjecture of Sidorenko and Erd\H{o}s-Simonovits states that if H is a bipartite graph then the random graph with edge density p has in expectation asymptotically the minimum number of copies of H over all graphs of the same order…

We present an algorithm for determining whether a bipartite graph $G$ is 2-chordal (formerly doubly chordal bipartite). At its core this algorithm is an extension of the existing efficient algorithm for determining whether a graph is…

Biregular bipartite graphs have been proven to have similar edge distributions to random bipartite graphs and thus have nice pseudorandomness and expansion properties. Thus it is quite desirable to find a biregular bipartite spanning…

This paper establishes sufficient conditions that force a graph to contain a bipartite subgraph with a given structural property. In particular, let $\beta$ be any of the following graph parameters: Hadwiger number, Haj\'{o}s number,…

Given a collection of graphs $\mathbf{G}=(G_1, \ldots, G_m)$ with the same vertex set, an $m$-edge graph $H\subset \cup_{i\in [m]}G_i$ is a transversal if there is a bijection $\phi:E(H)\to [m]$ such that $e\in E(G_{\phi(e)})$ for each…

A connected graph, on four or more vertices, is matching covered (aka 1-extendable) if every edge is present in some perfect matching. An ear decomposition theorem exists for bipartite matching covered graphs due to Hetyei. From the results…

The distinguishing number $D(G)$ of a graph $G$ is the least integer $d$ such that $G$ has a vertex labeling with $d$ labels that is preserved only by a trivial automorphism. The distinguishing chromatic number $\chi_{D}(G)$ of $G$ is…

Let $k$ be a positive integer and let $G$ be a graph with vertex set $V(G)$. A subset $D \subseteq V(G)$ is a $k$-dominating set if every vertex outside $D$ is adjacent to at least $k$ vertices in $D$. The $k$-domination number…

A beautiful conjecture of Erd\H{o}s-Simonovits and Sidorenko states that if H is a bipartite graph, then the random graph with edge density p has in expectation asymptotically the minimum number of copies of H over all graphs of the same…