Related papers: Weakly pancyclic vertices in dense nonbipartite gr…

We study the class of simple graphs $\mathcal{G}^*$ for which every pair of distinct odd cycles intersect in at most one edge. We give a structural characterization of the graphs in $\mathcal{G}^*$ and prove that every $G \in \mathcal{G}^*$…

It is shown that a hamiltonian $n/2$-regular bipartite graph $G$ of order $2n>8$ contains a cycle of length $2n-2$. Moreover, if such a cycle can be chosen to omit a pair of adjacent vertices, then $G$ is bipancyclic.

The vertex arboricity $a(G)$ of a graph $G$ is the minimum $k$ such that $V(G)$ can be partitioned into $k$ sets where each set induces a forest. For a planar graph $G$, it is known that $a(G)\leq 3$. In two recent papers, it was proved…

We consider the set of polygons all of whose sides are vertical or horizontal with fixed combinatorics (for example all the figure "L"s). We show that there is a dense G $\delta$ subset of such polygons such that for each polygon in this G…

Extremal problems for $3$-uniform hypergraphs are known to be very difficult and despite considerable effort the progress has been slow. We suggest a more systematic study of extremal problems in the context of quasirandom hypergraphs. We…

An $acyclic$ edge coloring of a graph is a proper edge coloring such that there are no bichromatic cycle s. The \emph{acyclic chromatic index} of a graph is the minimum number k such that there is an acyclic e dge coloring using k colors…

To find all the possible spectra of all real symmetric matrices whose off-diagonal pattern is prescribed by the adjacencies of a given graph $G$, the Strong Spectral Property turned out to be of crucial importance. In particular, we…

Clustering a signed graph means partitioning the vertices into sets ("clusters") so that every positive edge, and no negative edge, is within a cluster. Clustering is not always possible; the obstruction is circles with exactly one negative…

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…

A weakly distance-regular digraph is thick if its attached scheme is regular. In this paper, we show that each commutative thick weakly distance-regular digraph has a thick weakly distance-regular subdigraph such that the corresponding…

We study convex polyhedra in $\mathbb{R}\mathbb{P}^3$ with all their vertices on a sphere. We do not require, in particular, that the polyhedra lie in the interior of the sphere, hence the term "weakly inscribed". Such polyhedra can be…

Given $q$-uniform hypergraphs ($q$-graphs) $F,G$ and $H$, where $G$ is a spanning subgraph of $F$, $G$ is called weakly $H$-saturated in $F$ if the edges in $E(F)\setminus E(G)$ admit an ordering $e_1,\dots, e_k$ so that for all $i\in [k]$…

The {\em square} of a graph $G$, denoted $G^2$, has the same vertex set as $G$ and an edge between any two vertices at distance at most $2$ in $G$. Wegner (1977) conjectured that for a planar graph $G$, $\chi(G^2) \leq 7$ if $\Delta(G) =…

We prove that a strongly connected balanced bipartite digraph $D$ of order $2a$, $a\geq3$, satisfying $d(u)+d(v)\geq 3a$ for every pair of vertices $u,v$ with a common in-neighbour or a common out-neighbour, is either bipancyclic or a…

A graph $G$ on $m$ vertices is pancyclic if it contains cycles of length $l$, $3\leq l \leq m$ as subgraphs in $G$. The complete graph $K_{9}$ on 9 vertices with a cycle $C_{9}$ of length 9 deleted from $K_{9}$ is denoted by…

We prove that for any integer $k\geq 2$ and $\varepsilon>0$, there is an integer $\ell_0\geq 1$ such that any $k$-uniform hypergraph on $n$ vertices with minimum codegree at least $(1/2+\varepsilon)n$ has a fractional decomposition into…

A graph $G$ is $(c,t)$-sparse if for every pair of vertex subsets $A,B\subset V(G)$ with $|A|,|B|\geq t$, $e(A,B)\leq (1-c)|A||B|$. In this paper we prove that for every $c>0$ and integer $\ell$, there exists $C>1$ such that if an…

Let $G_w$ be a weighted graph. The number of the positive, negative and zero eigenvalues in the spectrum of $G_w$ are called positive inertia index, negative inertia index and nullity of $G_w$, and denoted by $i_{+}(G_w)$, $i_{-}(G_w)$,…

A well-known result of Mantel asserts that every $n$-vertex triangle-free graph $G$ has at most $\lfloor n^2/4 \rfloor$ edges. Moreover, Erd\H{o}s proved that if $G$ is further non-bipartite, then $e(G)\le \lfloor {(n-1)^2}/{4}\rfloor +1$.…

In this paper, we give a characterization of unicyclic graphs with diameter at most 4 which are A-vertex magic. Moreover, let G be a bicyclic graph of diameter 3, then G is group vertex magic if and only if G = M11(0, 0).