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The square $G^2$ of a graph $G$ is the graph defined on $V(G)$ such that two vertices $u$ and $v$ are adjacent in $G^2$ if the distance between $u$ and $v$ in $G$ is at most 2. Let $\chi(H)$ and $\chi_{\ell}(H)$ be the chromatic number and…

The distinguishing number (index) $D(G)$ ($D'(G)$) of a graph $G$ is the least integer $d$ such that $G$ has an vertex labeling (edge labeling) with $d$ labels that is preserved only by a trivial automorphism. The lexicographic product of…

We investigate the property of a spatial graph of having a leveled embedding and characterize the abstract graphs with this property. We show that all leveled embeddings are free and we compare leveled and paneled (also known as flat)…

List colouring is an influential and classic topic in graph theory. We initiate the study of a natural strengthening of this problem, where instead of one list-colouring, we seek many in parallel. Our explorations have uncovered a…

A graph $G$ of order $n>2$ is pancyclic if $G$ contains a cycle of length $l$ for each integer $l$ with $3 \leq l \leq n $ and it is called vertex-pancyclic if every vertex is contained in a cycle of length $l$ for every $3 \leq l \leq n $.…

In this paper we show that Leavitt path algebras of weighted graphs and Leavitt path algebras of separated graphs are intimately related. We prove that any Leavitt path algebra $L(E,\omega)$ of a row-finite vertex weighted graph…

Let $G$ be a graph with $p(G)$ pendant vertices and $q(G)$ quasi-pendant vertices. Denote by $m_{L(G)}(\lambda)$ the multiplicity of $\lambda$ as a Laplacian eigenvalue of $G$. Let $\overline{G}$ be the reduced graph of $G$, which can be…

In this paper we introduce the notion of $\Sigma$-colouring of a graph $G$: For given subsets $\Sigma(v)$ of neighbours of $v$, for every $v\in V(G)$, this is a proper colouring of the vertices of $G$ such that, in addition, vertices that…

A 2-edge-coloured graph $G$ is called {\bf locally complete} if for each vertex $v$, the vertices adjacent to $v$ through edges of the same colour induce a complete subgraph in $G$. Locally complete 2-edge-coloured graphs have nice…

A {\em packing coloring} of a graph $G$ is a mapping assigning a positive integer (a color) to every vertex of $G$ such that every two vertices of color $k$ are at distance at least $k+1$. The least number of colors needed for a packing…

We say that a graph $G$ is $(2,m)$-linked if, for any distinct vertices $a_1,\ldots, a_m, b_1,b_2$ in $G$, there exist vertex disjoint connected subgraphs $A,B$ of $G$ such that $\{a_1, \ldots, a_m\}$ is contained in $A$ and $\{b_1,b_2\}$…

An edge-coloring of a graph $G$ with colors $1,\ldots,t$ is an interval $t$-coloring if all colors are used, and the colors of edges incident to each vertex of $G$ are distinct and form an interval of integers. A graph $G$ is interval…

In this paper, we study the Hamiltonicity of graphs with large minimum degree. Firstly, we present some conditions for a simple graph to be Hamilton-connected and traceable from every vertex in terms of the spectral radius of the graph or…

A \emph{mixed interval graph} is an interval graph that has, for every pair of intersecting intervals, either an arc (directed arbitrarily) or an (undirected) edge. We are particularly interested in scenarios where edges and arcs are…

Answering a question raised by Dudek and Pra\l{}at, we show that if $pn\rightarrow \infty$, w.h.p.,~whenever $G=G(n,p)$ is $2$-coloured, there exists a monochromatic path of length $n(2/3+o(1))$. This result is optimal in the sense that…

In the last years, connection concepts such as rainbow connection and proper connection appeared in graph theory and obtained a lot of attention. In this paper, we investigate the loose edge-connection of graphs. A connected edge-coloured…

A vertex coloring of a graph $G$ is called a $2$-distance coloring if any two vertices at a distance at most $2$ from each other receive different colors. Recently, Bousquet et al. (Discrete Mathematics, 346(4), 113288, 2023) proved that…

A mapping $\alpha : V(G) \to V(H)$ from the vertex set of one graph $G$ to another graph $H$ is an isometric embedding if the shortest path distance between any two vertices in $G$ equals the distance between their images in $H$. Here, we…

A list assignment $L$ for a graph $G$ is an $(\ell,k)$-list assignment if $|L(v)|\geq \ell$ for each $v \in V(G)$ and $|L(u) \cap L(v)| \leq k$ for each $uv \in E(G)$. We say $G$ is $(\ell,k)$-choosable if it admits an $L$-colouring for…

List packing is a notion that was introduced in 2021 (by Cambie et al.). The list packing number of a graph $G$, denoted $\chi_{\ell}^*(G)$, is the least $k$ such that for any list assignment $L$ that assigns $k$ colors to each vertex of…