Related papers: Graph Operations and Upper Bounds on Graph Homomor…

The distinguishing number $\operatorname D(G)$ of a graph $G$ is the least cardinal $d$ such that $G$ has a labeling with $d$ labels which is only preserved by the trivial automorphism. We show that the distinguishing number of infinite,…

We present a new approach to showing that random graphs are nearly optimal expanders. This approach is based on recent deep results in combinatorial group theory. It applies to both regular and irregular random graphs. Let G be a random…

In this paper, we address problems related to parameters concerning edge mappings of graphs. The quantity $h(n,G)$ is defined to be the maximum number of edges in an $n$-vertex graph $H$ such that there exists a mapping $f: E(H)\rightarrow…

We present progress on three old conjectures about longest paths and cycles in graphs. The first pair of conjectures, due to Lov\'{a}sz from 1969 and Thomassen from 1978, respectively, states that all connected vertex-transitive graphs…

Given a multigraph $G$, the all-terminal reliability $R(G,p)$ is the probability that $G$ remains connected under percolation with parameter $p$. Fixing the number of vertices $n$ and edges $m$, we investigate which graphs maximize $R(G,p)$…

We describe an algorithm for the exhaustive generation of non-isomorphic graphs with a given number $k \ge 0$ of hamiltonian cycles, which is especially efficient for small $k$. Our main findings, combining applications of this algorithm…

For a positive integer $k$ and a graph $H$ on $k$ vertices, we are interested in the inducibility of $H$, denoted $\mathrm{ind}(H)$, which is defined as the maximum possible probability that choosing $k$ vertices uniformly at random from a…

We study the problem of computing the parity of the number of homomorphisms from an input graph $G$ to a fixed graph $H$. Faben and Jerrum [ToC'15] introduced an explicit criterion on the graph $H$ and conjectured that, if satisfied, the…

In this paper we are interested in the fine-grained complexity of deciding whether there is a homomorphism from an input graph $G$ to a fixed graph $H$ (the $H$-Coloring problem). The starting point is that these problems can be viewed as…

Let ${\rm dim}(G)$ and $D(G)$ respectively denote the metric dimension and the distinguishing number of a graph $G$. It is proved that $D(G) \le {\rm dim}(G)+1$ holds for every connected graph $G$. Among trees, exactly paths and stars…

Let $H$ be a fixed graph on $v$ vertices. For an $n$-vertex graph $G$ with $n$ divisible by $v$, an $H$-{\em factor} of $G$ is a collection of $n/v$ copies of $H$ whose vertex sets partition $V(G)$. In this paper we consider the threshold…

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…

Two graphs $G$ and $H$ are homomorphism indistinguishable over a family of graphs $\mathcal{F}$ if for all graphs $F \in \mathcal{F}$ the number of homomorphisms from $F$ to $G$ is equal to the number of homomorphism from $F$ to $H$. Many…

Given graphs $H_1,\ldots, H_k$, a graph $G$ is $(H_1,\ldots, H_k)$-free if there is a $k$-edge-colouring $\phi:E(G)\rightarrow [k]$ with no monochromatic copy of $H_i$ with edges of colour $i$ for each $i\in[k]$. Fix a function $f(n)$, the…

Let $G$ be a simple graph with order $n$, maximum degree $\D(G)$, minimum degree $\delta(G)$ and chromatic index $\chi'(G)$, respectively. A graph $G$ is called {\em $\D$-critical} if $\chi'(G)=\D(G)+1$ and $\chi'(H)\textless \chi'(G)$ for…

A collection of graphs is \textit{nearly disjoint} if every pair of them intersects in at most one vertex. We prove that if $G_1, \dots, G_m$ are nearly disjoint graphs of maximum degree at most $D$, then the following holds. For every…

A graph homomorphism is a vertex map which carries edges from a source graph to edges in a target graph. The instances of the Weighted Maximum H-Colourable Subgraph problem (MAX H-COL) are edge-weighted graphs G and the objective is to find…

It is well known [Lov\'asz, 67] that up to isomorphism a graph~$G$ is determined by the homomorphism counts $\hom(F, G)$, i.e., the number of homomorphisms from $F$ to $G$, where $F$ ranges over all graphs. Thus, in principle, we can answer…

A graph $G$ is said to have \textit{bandwidth} at most $b$, if there exists a labeling of the vertices by $1,2,..., n$, so that $|i - j| \leq b$ whenever $\{i,j\}$ is an edge of $G$. Recently, B\"{o}ttcher, Schacht, and Taraz verified a…

Let $\text{ch}(G)$ denote the choice number of a graph $G$ (also called "list chromatic number" or "choosability" of $G$). Noel, Reed, and Wu proved the conjecture of Ohba that $\text{ch}(G)=\chi(G)$ when $|V(G)|\le 2\chi(G)+1$. We extend…