Related papers: Sparse graphs are near-bipartite

For a bipartite graph G, let h(G) be the largest t such that either G or the bipartite complement of G contain K_{t,t}. For a class F of graphs, let h(F)= min {h(G): G\in F}. We say that a bipartite graph H is strongly acyclic if neither H…

The \emph{Antimagic Graph Conjecture} asserts that every connected graph $G = (V, E)$ except $K_2$ admits an edge labeling such that each label $1, 2, ..., |E|$ is used exactly once and the sums of the labels on all edges incident with a…

We prove that there is a constant $c >0$, such that whenever $p \ge n^{-c}$, with probability tending to 1 when $n$ goes to infinity, every maximum triangle-free subgraph of the random graph $G_{n,p}$ is bipartite. This answers a question…

A bipartite graph $G=(L,R;E)$ with at least one edge is said to be identifiable if for every vertex $v\in L$, the subgraph induced by its non-neighbors has a matching of cardinality $|L|-1$. An $\ell$-subgraph of $G$ is an induced subgraph…

Bipartite graphs are a fundamental concept in graph theory with diverse applications. A graph is bipartite iff it contains no odd cycles, a characteristic that has many implications in diverse fields ranging from matching problems to the…

In this paper we investigate the bipartite analogue of the strong Erdos-Hajnal property. We prove that for every forest $H$ and every $\tau>0$ there exists $\epsilon>0$, such that if $G$ has a bipartition $(A,B)$ and does not contain $H$ as…

Associated to a graph $G$ is a set $\mathcal{S}(G)$ of all real-valued symmetric matrices whose off-diagonal entries are nonzero precisely when the corresponding vertices of the graph are adjacent, and the diagonal entries are free to be…

We present evidence in support of a conjecture that a bipartite graph with at least five vertices in each part and |E(G)| \geq 4 |V(G)| - 17 is intrinsically knotted. We prove the conjecture for graphs that have exactly five or exactly six…

Let $S$ be a set of integers. A graph G is said to have the S-property if there exists an S-edge-weighting $w : E(G) \rightarrow S$ such that any two adjacent vertices have different sums of incident edge-weights. In this paper we…

Let $\alpha(G)$ denote the cardinality of a maximum independent set, while $\mu(G)$ be the size of a maximum matching in $G=\left( V,E\right) $. It is known that if $\alpha(G)+\mu(G)=\left\vert V\right\vert $, then $G$ is a…

An $(s,t)$-matching in a bipartite graph $G=(U,V,E)$ is a subset of the edges $F$ such that each component of $G[F]$ is a tree with at most $t$ edges and each vertex in $U$ has $s$ neighbours in $G[H]$. We give sharp conditions for a…

The parameter $q(G)$ of an $n$-vertex graph $G$ is the minimum number of distinct eigenvalues over the family of symmetric matrices described by $G$. We show that all $G$ with $e(\overline{G}) = |E(\overline{G})| \leq \lfloor n/2 \rfloor…

Let $G=(V,E)$ be a graph with the vertex-set $V$ and the edge-set $E$. Let $N(v)$ denote the set of neighbors of the vertex $v$ of $G.$ The graph $G$ is called $ irreducible $ whenever for every $v,w \in V$ if $v \neq w$, then $N(v)\neq…

Let $G$ be a group and $L(G)$ be the set of all subgroups of $G$. We introduce a bipartite graph $\mathcal{B}(G)$ on $G$ whose vertex set is the union of two sets $G \times G$ and $L(G)$, and two vertices $(a, b) \in G \times G$ and $H \in…

A bipartite covering of a (multi)graph $G$ is a collection of bipartite graphs, so that each edge of $G$ belongs to at least one of them. The capacity of the covering is the sum of the numbers of vertices of these bipartite graphs. In this…

A brick is a $3$-connected graph such that the graph obtained from it by deleting any two distinct vertices has a perfect matching. A brick $G$ is near-bipartite if it has a pair of edges $\alpha$ and $\beta$ such that $G-\{\alpha,\beta\}$…

Let $G$ be a bipartite graph and its adjacency matrix $\mathbb A$. If $G$ has a unique perfect matching, then $\mathbb A$ has an inverse $\mathbb A^{-1}$ which is a symmetric integral matrix, and hence the adjacency matrix of a multigraph.…

A graph is almost bipartite if it contains exactly one odd cycle, and it is Konig-Egervary if the sum of the independence number and the matching number equals the order of the graph. We introduce the class of Bipartite-Almost Bipartite…

A graph $G = (V, E)$ is word-representable, if there exists a word $w$ over the alphabet $V$ such that for letters $\{x,y\}\in V$, $x$ and $y$ alternate in $w$ if and only if $xy \in E$. A graph is co-bipartite if its complement is a…

Let $G_S$ be a graph with loops obtained from a graph $G$ of order $n$ and loops at $S \subseteq V(G)$. In this paper, we establish a neccesary and sufficient condition on the bipartititeness of a connected graph $G$ and the spectrum…