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We consider a weighted counting problem on matchings, denoted $\textrm{PrMatching}(\mathcal{G})$, on an arbitrary fixed graph family $\mathcal{G}$. The input consists of a graph $G\in \mathcal{G}$ and of rational probabilities of existence…

In a 1995 paper Richard Stanley defined $X_G$, the symmetric chromatic polynomial of a Graph $G=(V,E)$. He then conjectured that $X_G$ distinguishes trees; a conjecture which still remains open. $X_G$ can be represented as a certain…

We study the rainbow version of the graph commonness property: a graph $H$ is $r$-rainbow common if the number of rainbow copies of $H$ (where all edges have distinct colors) in an $r$-coloring of edges of $K_n$ is maximized asymptotically…

Given an edge-coloured graph, we say that a subgraph is rainbow if all of its edges have different colours. Let $\operatorname{ex}(n,H,$rainbow-$F)$ denote the maximal number of copies of $H$ that a properly edge-coloured graph on $n$…

A proper vertex coloring of a connected graph $G$ is called an odd coloring if, for every vertex $v$ in $G$, there exists a color that appears odd number of times in the open neighborhood of $v$. The minimum number of colors required to…

Given a graph $H$, we say that a graph $G$ is properly rainbow $H$-saturated if: (1) There is a proper edge colouring of $G$ containing no rainbow copy of $H$; (2) For every $e \notin E(G)$, every proper edge colouring of $G+e$ contains a…

The class of $\mathsf{Ga}$lled-$\mathsf{T}$ree $\mathsf{Ex}$plainable ($\mathsf{GaTEx}$) graphs has recently been discovered as a natural generalization of cographs. Cographs are precisely those graphs that can be uniquely represented by a…

For an undirected, simple, finite, connected graph $G$, we denote by $V(G)$ and $E(G)$ the sets of its vertices and edges, respectively. A function $\varphi:E(G)\rightarrow\{1,2,\ldots,t\}$ is called a proper edge $t$-coloring of a graph…

A graph is said to be \emph{total-colored} if all the edges and the vertices of the graph are colored. A total-colored graph is \emph{total-rainbow connected} if any two vertices of the graph are connected by a path whose edges and internal…

The Sombor index of a graph $G$ was recently introduced by Gutman from the geometric point of view, defined as $SO(G)=\sum_{uv\in E(G)}\sqrt{d(u)^2+d(v)^2}$, where $d(u)$ is the degree of a vertex $u$. For two real numbers $\alpha$ and…

In a vertex-colored graph $G = (V, E)$, a subset $S \subseteq V$ is said to be consistent if every vertex has a nearest neighbor in $S$ with the same color. The problem of computing a minimum cardinality consistent subset of a graph is…

A tree in an edge colored graph is said to be a rainbow tree if no two edges on the tree share the same color. Given two positive integers $k$, $\ell$ with $k\geq 3$, the \emph{$(k,\ell)$-rainbow index} $rx_{k,\ell}(G)$ of $G$ is the…

Let G be a combinatorial graph with vertices V and edges E. A proper coloring of G is an assignment of colors to the vertices such that no edge connects two vertices of the same color. These are the colorings considered in the famous Four…

We investigate the complexity of several fundamental polynomial-time solvable problems on graphs and on matrices, when the given instance has low treewidth; in the case of matrices, we consider the treewidth of the graph formed by non-zero…

We define an algorithm k which takes a connected graph G on a totally ordered vertex set and returns an increasing tree R (which is not necessarily a subtree of G). We characterize the set of graphs G such that k(G)=R. Because this set has…

A cactus graph is a graph in which any two cycles are edge-disjoint. We present a constructive proof of the fact that any plane graph $G$ contains a cactus subgraph $C$ where $C$ contains at least a $\frac{1}{6}$ fraction of the triangular…

An edge-colored multigraph $G$ is rainbow connected if every pair of vertices is joined by at least one rainbow path, i.e., a path where no two edges are of the same color. In the context of multilayered networks we introduce the notion of…

A spanning tree of a properly edge-colored complete graph, $K_n$, is rainbow provided that each of its edges receives a distinct color. In 1996, Brualdi and Hollingsworth conjectured that if $K_{2m}$ is properly $(2m-1)$-edge-colored, then…

Rainbow connection number rc(G) of a connected graph G is the minimum number of colours needed to colour the edges of G, so that every pair of vertices is connected by at least one path in which no two edges are coloured the same. In this…

A path in an edge-colored graph $G$, where adjacent edges may be colored the same, is a rainbow path if every two edges of it receive distinct colors. The rainbow connection number of a connected graph $G$, denoted by $rc(G)$, is the…