Related papers: Computational complexity of the Weisfeiler-Leman d…
Learning to optimize is a rapidly growing area that aims to solve optimization problems or improve existing optimization algorithms using machine learning (ML). In particular, the graph neural network (GNN) is considered a suitable ML model…
Given an $n$-vertex graph $G$ and two positive integers $d,k \in \mathbb{N}$, the ($d,kn$)-differential coloring problem asks for a coloring of the vertices of $G$ (if one exists) with distinct numbers from 1 to $kn$ (treated as…
In a bounded max-coloring of a vertex/edge weighted graph, each color class is of cardinality at most $b$ and of weight equal to the weight of the heaviest vertex/edge in this class. The bounded max-vertex/edge-coloring problems ask for…
The paper considers the NP-hard graph vertex coloring problem, which differs from traditional problems in which it is required to color vertices with a given (or minimal) number of colors so that adjacent vertices have different colors. In…
A set of vertices $W$ in a graph $G$ is called resolving if for any two distinct $x,y\in V(G)$, there is $v\in W$ such that ${\rm dist}_G(v,x)\neq{\rm dist}_G(v,y)$, where ${\rm dist}_G(u,v)$ denotes the length of a shortest path between…
An assignment of numbers to the vertices of graph G is closed distinguishing if for any two adjacent vertices v and u the sum of labels of the vertices in the closed neighborhood of the vertex v differs from the sum of labels of the…
A square coloring of a graph $G$ is a coloring of the square $G^2$ of $G$, that is, a coloring of the vertices of $G$ such that any two vertices that are at distance at most $2$ in $G$ receive different colors. We investigate the complexity…
A set of vertices $W$ resolves a graph $G$ if every vertex is uniquely determined by its vector of distances to the vertices in $W$. A metric dimension of $G$ is the minimum cardinality of a resolving set of $G$. A bipartite graph G(n,n) is…
For a proper vertex coloring $c$ of a graph $G$, let $\varphi_c(G)$ denote the maximum, over all induced subgraphs $H$ of $G$, the difference between the chromatic number $\chi(H)$ and the number of colors used by $c$ to color $H$. We…
The problem of graph isomorphism is an important but challenging problem in the field of graph analysis, for example: analyzing the similarity of two chemical molecules, or studying the expressive ability of graph neural networks. WL test…
The expressive power of graph neural networks (GNNs) is typically understood through their correspondence with graph isomorphism tests such as the Weisfeiler-Leman (WL) hierarchy. While more expressive GNNs can distinguish a richer set of…
List k-Coloring (Li k-Col) is the decision problem asking if a given graph admits a proper coloring compatible with a given list assignment to its vertices with colors in {1,2,..,k}. The problem is known to be NP-hard even for k=3 within…
One of the fundamental open problems in the area of distributed graph algorithms is the question of whether randomization is needed for efficient symmetry breaking. While there are fast, $\text{poly}\log n$-time randomized distributed…
A graph $G$ is $k$-vertex-critical if $G$ has chromatic number $k$ but every proper induced subgraph of $G$ has chromatic number less than $k$. The study of $k$-vertex-critical graphs for graph classes is an important topic in algorithmic…
Two graphs are homomorphism indistinguishable over a graph class $\mathcal{F}$, denoted by $G \equiv_{\mathcal{F}} H$, if $\operatorname{hom}(F,G) = \operatorname{hom}(F,H)$ for all $F \in \mathcal{F}$ where $\operatorname{hom}(F,G)$…
Subgraph Isomorphism is a very basic graph problem, where given two graphs $G$ and $H$ one is to check whether $G$ is a subgraph of $H$. Despite its simple definition, the Subgraph Isomorphism problem turns out to be very broad, as it…
An injective $k$-edge-coloring of a graph $G$ is an assignment of colors, i.e. integers in $\{1, \ldots , k\}$, to the edges of $G$ such that any two edges each incident with one distinct endpoint of a third edge, receive distinct colors.…
In this paper we combine many of the standard and more recent algebraic techniques for testing isomorphism of finite groups (GpI) with combinatorial techniques that have typically been applied to Graph Isomorphism. In particular, we show…
A well-studied coloring problem is to assign colors to the edges of a graph $G$ so that, for every pair of vertices, all edges of at least one shortest path between them receive different colors. The minimum number of colors necessary in…
It is well-known that the graph isomorphism problem can be posed as an equivalent problem of determining whether an auxiliary graph structure contains a clique of specific order. However, the algorithms that have been developed so far for…