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The $k$-Colouring problem is to decide if the vertices of a graph can be coloured with at most $k$ colours for a fixed integer $k$ such that no two adjacent vertices are coloured alike. If each vertex u must be assigned a colour from a…

We survey results concerning reconfigurations of colourings and dominating sets in graphs. The vertices of the $k$-colouring graph $\mathcal{C}_{k}(G)$ of a graph $G$ correspond to the proper $k$-colourings of a graph $G$, with two…

For $k \geqslant 0$, we define a simple topological graph $G$ (that is, a graph drawn in the plane such that every pair of edges intersect at most once, including endpoints) to be $k$-matching-planar if for every edge $e \in E(G)$, every…

As an extension of the Brooks theorem, Catlin in 1979 showed that if $H$ is neither an odd cycle nor a complete graph with maximum degree $\Delta(H)$, then $H$ has a vertex $\Delta(H)$-coloring such that one of the color classes is a…

For graphs $G$ and $H$, an {\em $H$-colouring} of $G$ (or {\em homomorphism} from $G$ to $H$) is a function from the vertices of $G$ to the vertices of $H$ that preserves adjacency. $H$-colourings generalize such graph theory notions as…

Lov\'asz (1967) showed that two graphs $G$ and $H$ are isomorphic if and only if they are homomorphism indistinguishable over the class of all graphs, i.e. for every graph $F$, the number of homomorphisms from $F$ to $G$ equals the number…

Given a graph $G$ and a target graph $H$, an $H$-coloring of $G$ is an adjacency-preserving vertex map from $G$ to $H$. The number of $H$-colorings of $G$, $\hom(G,H)$, has been studied for many classes of $G$ and $H$. In particular,…

Since the seminal result of Karger, Motwani, and Sudan, algorithms for approximate 3-coloring have primarily centered around SDP-based rounding. However, it is likely that important combinatorial or algebraic insights are needed in order to…

The criteria for determining graph isomorphism are crucial for solving graph isomorphism problems. The necessary condition is that two isomorphic graphs possess invariants, but their function can only be used to filtrate and subdivide…

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…

For any positive integer $k$, the reconfiguration graph for all $k$-colorings of a graph $G$, denoted by $\mathcal{R}_k(G)$, is the graph where vertices represent the $k$-colorings of $G$, and two $k$-colorings are joined by an edge if they…

A colouring of a graph $G=(V,E)$ is a function $c: V\rightarrow\{1,2,\ldots \}$ such that $c(u)\neq c(v)$ for every $uv\in E$. A $k$-regular list assignment of $G$ is a function $L$ with domain $V$ such that for every $u\in V$, $L(u)$ is a…

Assume $k$ is a positive integer, $\lambda=\{k_1, k_2, \ldots, k_q\}$ is a partition of $k$ and $G$ is a graph. A $\lambda$-list assignment of $G$ is a $k$-list assignment $L$ of $G$ such that the colour set $\cup_{v\in V(G)}L(v)$ can be…

Graph Isomorphism is one of the classical problems of graph theory for which no deterministic polynomial-time algorithm is currently known, but has been neither proven to be NP-complete. Several heuristic algorithms have been proposed to…

Let $\sigma$ be a partition of the positive integer $r$. A $\sigma$-hypergraph $H=H(n,r,q|\sigma)$ is an $r$-uniform hypergraph on $nq$ vertices which are partitioned into $n$ classes $V_1, V_2, \ldots, V_n$ each containing $q$ vertices. An…

Graph coloring problems are a central topic of study in the theory of algorithms. We study the problem of partially coloring partially colorable graphs. For $\alpha \leq 1$ and $k \in \mathbb{Z}^+$, we say that a graph $G=(V,E)$ is…

Representational limits of message-passing graph neural networks (MP-GNNs), e.g., in terms of the Weisfeiler-Leman (WL) test for isomorphism, are well understood. Augmenting these graph models with topological features via persistent…

Graph product structure theory expresses certain graphs as subgraphs of the strong product of much simpler graphs. In particular, an elegant formulation for the corresponding structural theorems involves the strong product of a path and of…

For graphs $G$ and $H$, a \emph{homomorphism} from $G$ to $H$ is an edge-preserving mapping from the vertex set of $G$ to the vertex set of $H$. For a fixed graph $H$, by \textsc{Hom($H$)} we denote the computational problem which asks…

Differential Privacy is the gold standard in privacy-preserving data analysis. This paper addresses the challenge of producing a differentially edge-private vertex coloring. In this paper, we present two novel algorithms to approach this…