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A vertex of a graph is bisimplicial if the set of its neighbors is the union of two cliques; a graph is quasi-line if every vertex is bisimplicial. A recent result of Chudnovsky and Seymour asserts that every non-empty even-hole-free graph…

We consider three probability measures on subsets of edges of a given finite graph $G$, namely those which govern, respectively, a uniform forest, a uniform spanning tree, and a uniform connected subgraph. A conjecture concerning the…

A conjecture of Barnette states that every 3-connected cubic bipartite plane graph has a Hamilton cycle, which is equivalent to the statement that every simple even plane triangulation admits a partition of its vertex set into two subsets…

The biclique partition number of a graph $G= (V,E)$, denoted $bp(G)$, is the minimum number of pairwise edge disjoint complete bipartite subgraphs of $G$ so that each edge of $G$ belongs to exactly one of them. It is easy to see that $…

Bollob\'{a}s and Nikiforov (J. Combin. Theory Ser. B. 97 (2007) 859-865) conjectured that for a graph $G$ with $e(G)$ edges and the clique number $\omega(G)$, then $ \lambda_{1}^{2}+\lambda_{2}^{2}\leq…

The primary objective of this paper is to investigate the notions of geometric and sequential convexity within a graph-theoretic framework, with the aim of examining various structural properties and exploring the connection between these…

The $W_v$-Path Conjecture due to Klee and Wolfe states that any two vertices of a simple polytope can be joined by a path that does not revisit any facet. This is equivalent to the well-known Hirsch Conjecture. Klee proved that the…

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…

Two graphs $G_1$ and $G_2$ on $n$ vertices are said to pack if there exist injective mappings of their vertex sets into $[n]$ such that the images of their edge sets are disjoint. A longstanding conjecture due to Bollob\'as and Eldridge…

For distinct vertices $u,v$ in a graph $G$, let $\kappa_G(u,v)$ denote the maximum number of internally disjoint $u$-$v$ paths in $G$. Then, $\kappa_G(u,v) \leq \min\{ \mbox{deg}_G(u), \mbox{deg}_G(v) \}$. If equality is attained for every…

Let $\hom(H,G)$ denote the number of homomorphisms from a graph $H$ to a graph $G$. Sidorenko's conjecture asserts that for any bipartite graph $H$, and a graph $G$ we have $$\hom(H,G)\geq…

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 cubical polytope is a polytope with all its facets being combinatorially equivalent to cubes. The paper is concerned with the linkedness of the graphs of cubical polytopes. A graph with at least $2k$ vertices is \textit{$k$-linked} if,…

In this paper, we study the Reconstruction Conjecture for finite simple graphs. Let $\Gamma$ and $\Gamma'$ be finite simple graphs with at least three vertices such that there exists a bijective map $f:V(\Gamma) \rightarrow V(\Gamma')$ and…

We prove that there exist bipartite Ramanujan graphs of every degree and every number of vertices. The proof is based on analyzing the expected characteristic polynomial of a union of random perfect matchings, and involves three…

A $k$-uniform hypergraph $G=(V,E)$ is called odd-bipartite ([5]), if $k$ is even and there exists some proper subset $V_1$ of $V$ such that each edge of $G$ contains odd number of vertices in $V_1$. Odd-bipartite hypergraphs are…

The class of cographs is one of the most well-known graph classes, which is also known to be equivalent to the class of $P_4$-free graphs. We show that Mader's conjecture is true if we restrict ourselves to cographs, that is, for any tree…

The Brouwer conjecture (BC) in spectral graph theory claims that the sum of the largest k Kirchhoff eigenvalues of a graph are bounded above by the number m of edges plus k(k+1)/2. We show that (BC) holds for all graphs with n vertices if n…

For an integer $k\ge 2$, let $G$ be a graph with $m$ edges and without cycles of length $2k$. The pivotal Alon-Krivelevich-Sudakov Theorem on Max-Cuts states that $G$ has a bipartite subgraph with at least $m/2+\Omega(m^{(2k+1)/(2k+2)})$…

A k-tree is either a complete graph on (k+1) vertices or given a k-tree G' with n vertices, a k-tree G with (n+1) vertices can be constructed by introducing a new vertex v and picking a k-clique Q in G' and then joining each vertex u in Q.…