Related papers: The antimagic orientation problems for graphs obta…

We consider the single-conflict coloring problem, a graph coloring problem in which each edge of a graph receives a forbidden ordered color pair. The task is to find a vertex coloring such that no two adjacent vertices receive a pair of…

Let $n$ be sufficiently large and suppose that $G$ is a digraph on $n$ vertices where every vertex has in- and outdegree at least $n/2$. We show that $G$ contains every orientation of a Hamilton cycle except, possibly, the antidirected one.…

The zero forcing number of a graph $G$, denoted by $Z(G)$, is the minimum cardinality of a set $S$ of black vertices (where vertices in $V(G)\setminus S$ are colored white) such that $V(G)$ is turned black after finitely many applications…

Graph labeling is a technique that assigns unique labels or weights to the vertices or edges of a graph, often used to analyze and solve various graph-related problems. There are few methods with certain limitations conducted by researchers…

For every $n\in\mathbb N$ we construct a finite graph $G$ such that every orientation $\vec G$ of $G$ contains an isometric copy of any oriented tree on $n$ vertices, and evaluate the smallest possible cardinality of $G$. On the other hand,…

Consider an undirected graph $G=(V,E)$. A subgraph of $G$ is a subset of its edges, whilst an orientation of $G$ is an assignment of a direction to each edge. Provided with an integer circulation-demand $d:V\to \mathbb{Z}$, we show an…

Lov\'{a}sz conjectured that every connected vertex-transitive graph contains a hamilton path in 1970. First we reveal the structure of connected vertex-transitive graphs with an odd number of vertices. Then we prove that every connected…

The cycles are the only $2$-connected graphs in which any two nonadjacent vertices form a vertex cut. We generalize this fact by proving that for every integer $k\ge 3$ there exists a unique graph $G$ satisfying the following conditions:…

A graph is an opposition graph, respectively, a coalition graph, if it admits an acyclic orientation which puts the two end-edges of every chordless 4-vertex path in opposition, respectively, in the same direction. Opposition and coalition…

Perfect graphs form one of the distinguished classes of finite simple graphs. In 2006, Chudnovsky, Robertson, Seymour and Thomas proved that a graph is perfect if and only if it has no odd holes and no odd antiholes as induced subgraphs,…

A graph $H$ is common if its Ramsey multiplicity, i.e., the minimum number of monochromatic copies of $H$ contained in any $2$-edge-coloring of $K_n$, is asymptotically the same as the number of monochromatic copies in the random…

We prove that every recurrent graph $G$ quasi-isometric to $\mathbb{R}$ admits an essentially unique Lipschitz harmonic function $h$. If $G$ is vertex-transitive, then the action of $Aut(G)$ preserves $\partial h$ up to a sign, a fact that…

In 2004, Karo\'nski, \L uczak and Thomason proposed $1$-$2$-$3$-conjecture: For every nice graph $G$ there is an edge weighting function $ w:E(G)\rightarrow\{1,2,3\} $ such that the induced vertex coloring is proper. After that, the total…

For two graphs $G$ and $H$, a mapping $f\colon E(G) \to E(H)$ is an $H$-coloring of $G$, if it is a proper edge-coloring and for every $v \in V(G)$ there exists a vertex $u \in V(H)$ with $f(\partial_G(v))=\partial_H(u)$. Motivated by the…

Lov\'asz conjectured that every connected 4-regular planar graph G admits a realization as a system of circles, i.e., it can be drawn on the plane utilizing a set of circles, such that the vertices of G correspond to the intersection and…

A drawing of a graph is said to be a {\em straight-line drawing} if the vertices of $G$ are represented by distinct points in the plane and every edge is represented by a straight-line segment connecting the corresponding pair of vertices…

Given an edge-colored graph $G$, we denote the number of colors as $c(G)$, and the number of edges as $e(G)$. An edge-colored graph is rainbow if no two edges share the same color. A proper $mK_3$ is a vertex disjoint union of $m$ rainbow…

The mean color number of an $n$-vertex graph $G$, denoted by $\mu(G)$, is the average number of colors used in all proper $n$-colorings of $G$. For any graph $G$ and a vertex $w$ in $G$, Dong (2003) conjectured that if $H$ is a graph…

A magic labelling of a graph $G$ with magic sum $s$ is a labelling of the edges of $G$ by nonnegative integers such that for each vertex $v\in V$, the sum of labels of all edges incident to $v$ is equal to the same number $s$. Stanley gave…

For unsigned graphs G and H, the characteristic polynomial of different graph matrices for edge corona, subdivision vertex neighbourhood corona and subdivision edge neighbourhood corona has already been studied using the concept of coronal.…