Related papers: Enumerating independent sets in Abelian Cayley gra…

We give constructive proofs for the existence of uniquely hamiltonian graphs for various sets of degrees. We give constructions for all sets with minimum 2 (a trivial case added for completeness), all sets with minimum 3 that contain an…

For a finite simplicial graph $\Gamma$, let $A(\Gamma)$ denote the right-angled Artin group on $\Gamma$. Recently Kim and Koberda introduced the extension graph $\Gamma^e$ for $\Gamma$, and established the Extension Graph Theorem: for…

By definition, the sharp packing index $\ind_P^\sharp(A)$ of a subset $A$ of an abelian group $G$ is the smallest cardinal $\kappa$ such that for any subset $B\subset G$ of size $|B|\ge\kappa$ the family $\{b+A:b\in B\}$ is not disjoint. We…

An $L(2,1)$-labelling of a finite graph $\Gamma$ is a function that assigns integer values to the vertices $V(\Gamma)$ of $\Gamma$ (colouring of $V(\Gamma)$ by ${\mathbb{Z}}$) so that the absolute difference of two such values is at least…

In this paper we introduce a Cayley-type graph for group-subgroup pairs and present some elementary properties of such graphs, including connectedness, their degree and partition structure, and vertex-transitivity. We relate these…

Alon, Krivelevich, and Sudakov conjectured in 1999 that every $F$-free graph of maximum degree at most $\Delta$ has chromatic number $O(\Delta / \log \Delta)$. This was previously known only for almost bipartite graphs, that is, for…

Counting the independent sets of a graph is a classical #P-complete problem, even in the bipartite case. We give an exponential-time approximation scheme for this problem which is faster than the best known algorithm for the exact problem.…

We give bounds on the L(2,1)-labeling number of a simple graph in terms of its order and its maximum degree. We also describe an infinite class of graphs of which the elements have the highest L(2,1)-labeling numbers in terms of their…

In this short survey article, we present an elementary, yet quite powerful, method of enumerating independent sets in graphs. This method was first employed more than three decades ago by Kleitman and Winston and has subsequently been used…

A perfect code in a graph $\Gamma = (V, E)$ is a subset $C$ of $V$ such that no two vertices in $C$ are adjacent and every vertex in $V \setminus C$ is adjacent to exactly one vertex in $C$. A total perfect code in $\Gamma$ is a subset $C$…

We construct a polynomial-time algorithm which given a graph $\Gamma$ finds the full set of non-equivalent Cayley representations of $\Gamma$ over the group $D\cong C_p\times C_{p^k}$, where $p\in\{2,3\}$ and $k\geq 1$. This result implies…

In this paper, we propose the following conjecture which generalizes a theorem proved by Huang [Hua19] in his recent breakthrough proof of the sensitivity conjecture. We conjecture that for any Cayley graph $X = \Gamma(G,S)$ on a group $G$…

Let $G$ denote a finite abelian group with identity 1 and let $S$ denote an inverse-closed subset of $G \setminus {1}$, which generates $G$ and for which there exists $s \in S$, such that $\la S \setminus \{s,s^{-1}\} \ra \ne G$. In this…

We fully disprove a conjecture of Haythorpe on the minimum number of hamiltonian cycles in regular hamiltonian graphs, thereby extending a result of Zamfirescu, as well as correct and complement Haythorpe's computational enumerative results…

Consider the family of all finite graphs with maximum degree $\Delta(G)<d$ and matching number $\nu(G)<m$. In this paper we give a new proof to obtain the exact upper bound for the number of edges in such graphs and also characterize all…

We determine the maximum number of maximal independent sets of arbitrary graphs in terms of their covering numbers and we completely characterize the extremal graphs. As an application, we give a similar result for K\"onig-Egerv\'ary graphs…

A maximal independent set in a graph $G$ is an independent set that cannot be extended to a larger independent set by adding any vertex from $G$. This paper investigates the problem of determining the maximum number of maximal independent…

We prove an upper bound for the independence number of a graph in terms of the largest Laplacian eigenvalue, and of a certain induced subgraph. Our bound is a refinement of a well-known Hoffman-type bound.

A graph is called integral if all its eigenvalues are integers. A Cayley graph is called normal if its connection set is a union of conjugacy classes. We show that a non-empty integral normal Cayley graph for a group of odd order has an odd…

We construct an infinite family of connected, 2-generated Cayley digraphs Cay(G;a,b) that do not have hamiltonian paths, such that the orders of the generators a and b are arbitrarily large. We also prove that if G is any finite group with…