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Given a graph $X$ with a Hamilton cycle $C$, the {\em compression factor $\kappa(X,C)$ of $C$} is the order of the largest cyclic subgroup of $\operatorname{Aut}(C)\cap\operatorname{Aut}(X)$, and the {\em Hamilton compression $\kappa(X)$ of $X$ } is the maximum of $\kappa(X,C)$ where $C$ runs over all Hamilton cycles in $X$. Generalizing the well-known open problem regarding the existence of vertex-transitive graphs without Hamilton paths/cycles, it was asked by Gregor, Merino and M\"utze in [``The Hamilton compression of highly symmetric graphs'', {\em arXiv preprint} arXiv: 2205.08126v1 (2022)] whether for every positive integer $k$ there exists infinitely many vertex-transitive graphs (Cayley graphs) with Hamilton compression equal to $k$. Since an infinite family of Cayley graphs with Hamilton compression equal to $1$ was given there, the question is completely resolved in this paper in the case of Cayley graphs with a construction of Cayley graphs of semidirect products $\mathbb{Z}_p\rtimes\mathbb{Z}_k$ where $p$ is a prime and $k \geq 2$ a divisor of $p-1$. Further, infinite families of non-Cayley vertex-transitive graphs with Hamilton compression equal to $1$ are given. All of these graphs being metacirculants, some additional results on Hamilton compression of metacirculants of specific orders are also given., Comment: 11 pages

For a permutation group $G$ acting on a set $V$, a subset $\mathcal{F}$ of $G$ is said to be an intersecting set if for every pair of elements $g,h\in \mathcal{F}$ there exists $v \in V$ such that $g(v) = h(v)$. The intersection density $\rho(G)$ of a transitive permutation group $G$ is the maximum value of the quotient $|\mathcal{F}|/|G_v|$ where $G_v$ is a stabilizer of a point $v\in V$ and $\mathcal{F}$ runs over all intersecting sets in $G$. If $G_v$ is a largest intersecting set in $G$ then $G$ is said to have the Erd\H{o}s-Ko-Rado (EKR)-property. This paper is devoted to the study of transitive permutation groups, with point stabilizers of prime order with a special emphasis given to orders 2 and 3, which do not have the EKR-property. Among other, constructions of infinite family of transitive permutation groups having point stabilizer of order $3$ with intersection density $4/3$ and of infinite families of transitive permutation groups having point stabilizer of order $3$ with arbitrarily large intersection density are given., Comment: 14 pages

For a permutation group $G$ acting on a set $V$, a subset $I$ of $G$ is said to be an intersecting set if for every pair of elements $g,h\in I$ there exists $v \in V$ such that $g(v) = h(v)$. The intersection density $\rho(G)$ of a transitive permutation group $G$ is the maximum value of the quotient $|I|/|G_v|$ where $G_v$ is a stabilizer of a point $v\in V$ and $I$ runs over all intersecting sets in $G$. If $G_v$ is the largest intersecting set in $G$ then $G$ is said to have the Erd\H{o}s-Ko-Rado (EKR)-property, and moreover, $G$ has the strict-EKR-property if every intersecting set of maximum size in $G$ is a coset of a point stabilizer. Intersecting sets in $G$ coincide with independent sets in the so-called derangement graph $\Gamma_G$, defined as the Cayley graph on $G$ with connection set consisting of all derangements, that is, fixed-point free elements of $G$. In this paper a conjecture regarding the existence of transitive permutation groups whose derangement graphs are complete multipartite graphs, posed by Meagher, Razafimahatratra and Spiga in [J.Combin. Theory Ser. A 180 (2021), 105390], is proved. The proof uses direct product of groups. Questions regarding maximum intersecting sets in direct and wreath products of groups and the (strict)-EKR-property of these group products are also investigated. In addition, some errors appearing in the literature on this topic are corrected., Comment: 13 pages

Two elements $g$ and $h$ of a permutation group $G$ acting on a set $V$ are said to be intersecting if $g(v) = h(v)$ for some $v \in V$. More generally, a subset ${\cal F}$ of $G$ is an intersecting set if every pair of elements of ${\cal F}$ is intersecting. The intersection density $\rho(G)$ of a transitive permutation group $G$ is the maximum value of the quotient $|{\cal F}|/|G_v|$ where $G_v$ is a stabilizer of $v\in V$ and ${\cal F}$ runs over all intersecting sets in $G$. Intersection densities of transitive groups of degree $pq$, where $p>q$ are odd primes, is considered. In particular, the conjecture that the intersection density of every such group is equal to $1$ (posed in [ J.~Combin. Theory, Ser. A 180 (2021), 105390]) is disproved by constructing a family of imprimitive permutation groups of degree $pq$ (with blocks of size $q$), where $p=(q^k-1)/(q-1)$, whose intersection density is equal to $q$. The construction depends heavily on certain equidistant cyclic codes $[p,k]_q$ over the field $\mathbb{F}_q$ whose codewords have Hamming weight strictly smaller than $p$., Comment: 8 pages

Two elements $g$ and $h$ of a permutation group $G$ acting on a set $V$ are said to be intersecting if $g(v) = h(v)$ for some $v \in V$. More generally, a subset ${\cal F}$ of $G$ is an intersecting set if every pair of elements of ${\cal F}$ is intersecting. The intersection density $\rho(G)$ of a transitive permutation group $G$ is the maximum value of the quotient $|{\cal F}|/|G_v|$ where ${\cal F}$ runs over all intersecting sets in $G$ and $G_v$ is a stabilizer of $v\in V$. In this paper the intersection density of transitive groups of degree twice a prime is determined, and proved to be either $1$ or $2$. In addition, it is proved that the intersection density of transitive groups of prime power degree is $1$., Comment: 10 pages

A step forward is made in a long standing Lov\'{a}sz's problem regarding hamiltonicity of vertex-transitive graphs by showing that every connected vertex-transitive graph of order a product of two primes, other than the Petersen graph, contains a Hamilton cycle. Essential tools used in the proof range from classical results on existence of Hamilton cycles, such as Chv\'atal's theorem and Jackson's theorem, to certain results on polynomial representations of quadratic residues at primitive roots in finite fields., Comment: 50 pages, 5 figures, 7 tables

When dealing with symmetry properties of mathematical objects, one of the fundamental questions is to determine their full automorphism group. In this paper this question is considered in the context of even/odd permutations dichotomy. More precisely: when is it that existence of automorphisms acting as even permutations on the vertex set of a graph, called {\em even automorphisms}, forces existence of automorphisms that act as odd permutations, called {\em odd automorphisms}. As a first step towards resolving the above question, a complete information on existence of odd automorphisms in cubic symmetric graphs is given.

It was proved in [Y.-Q. Feng, C. H. Li and J.-X. Zhou, Symmetric cubic graphs with solvable automorphism groups, {\em European J. Combin.} {\bf 45} (2015), 1-11] that a cubic symmetric graph with a solvable automorphism group is either a Cayley graph or a $2$-regular graph of type $2^2$, that is, a graph with no automorphism of order $2$ interchanging two adjacent vertices. In this paper an infinite family of non-Cayley cubic $2$-regular graphs of type $2^2$ with a solvable automorphism group is constructed. The smallest graph in this family has order 6174., Comment: 8 pages

In this paper it is shown that every finite cyclic group satisfies the CI-property for the class of balanced configurations.

A graph is said to be cyclic $k$-edge-connected, if at least $k$ edges must be removed to disconnect it into two components, each containing a cycle. Such a set of $k$ edges is called a cyclic-$k$-edge cutset and it is called a trivial cyclic-$k$-edge cutset if at least one of the resulting two components induces a single $k$-cycle. It is known that fullerenes, that is, 3-connected cubic planar graphs all of whose faces are pentagons and hexagons, are cyclic 5-edge-connected. In this article it is shown that a fullerene $F$ containing a nontrivial cyclic-5-edge cutset admits two antipodal pentacaps, that is, two antipodal pentagonal faces whose neighboring faces are also pentagonal. Moreover, it is shown that $F$ has a Hamilton cycle, and as a consequence at least $15\cdot 2^{\lfloor \frac{n}{20}\rfloor}$ perfect matchings, where $n$ is the order of $F$., Comment: 11 pages, 9 figures

Following Alspach and Parsons, a {\em metacirculant graph} is a graph admitting a transitive group generated by two automorphisms $\rho$ and $\sigma$, where $\rho$ is $(m,n)$-semiregular for some integers $m \geq 1$, $n \geq 2$, and where $\sigma$ normalizes $\rho$, cyclically permuting the orbits of $\rho$ in such a way that $\sigma^m$ has at least one fixed vertex. A {\em half-arc-transitive graph} is a vertex- and edge- but not arc-transitive graph. In this article quartic half-arc-transitive metacirculants are explored and their connection to the so called tightly attached quartic half-arc-transitive graphs is explored. It is shown that there are three essentially different possibilities for a quartic half-arc-transitive metacirculant which is not tightly attached to exist. These graphs are extensively studied and some infinite families of such graphs are constructed., Comment: 31 pages, 2 figures

A regular covering projection $\p\colon \tX \to X$ of connected graphs is $G$-admissible if $G$ lifts along $\p$. Denote by $\tG$ the lifted group, and let $\CT(\p)$ be the group of covering transformations. The projection is called $G$-split whenever the extension $\CT(\p) \to \tG \to G$ splits. In this paper, split 2-covers are considered. Supposing that $G$ is transitive on $X$, a $G$-split cover is said to be $G$-split-transitive if all complements $\bG \cong G$ of $\CT(\p)$ within $\tG$ are transitive on $\tX$; it is said to be $G$-split-sectional whenever for each complement $\bG$ there exists a $\bG$-invariant section of $\p$; and it is called $G$-split-mixed otherwise. It is shown, when $G$ is an arc-transitive group, split-sectional and split-mixed 2-covers lead to canonical double covers. For cubic symmetric graphs split 2-cover are necessarily cannonical double covers when $G$ is 1- or 4-regular. In all other cases, that is, if $G$ is $s$-regular, $s=2,3$ or 5, a necessary and sufficient condition for the existence of a transitive complement $\bG$ is given, and an infinite family of split-transitive 2-covers based on the alternating groups of the form $A_{12k+10}$ is constructed. Finally, chains of consecutive 2-covers, along which an arc-transitive group $G$ has successive lifts, are also considered. It is proved that in such a chain, at most two projections can be split. Further, it is shown that, in the context of cubic symmetric graphs, if exactly two of them are split, then one is split-transitive and the other one is either split-sectional or split-mixed., Comment: 18 pages, 3 figures

The following problem is considered: if $H$ is a semiregular abelian subgroup of a transitive permutation group $G$ acting on a finite set $X$, find conditions for (non) existence of $G$-invariant partitions of $X$. Conditions presented in this paper are derived by studying spectral properties of associated $G$-invariant digraphs. As an essential tool, irreducible complex characters of $H$ are used. Questions of this kind arise naturally when classifying combinatorial objects which enjoy a certain degree of symmetry. As an illustration, a new and short proof of an old result of Frucht, Graver and Watkins ({\it Proc. Camb. Phil. Soc.}, {\bf 70} (1971), 211-218) classifying edge-transitive generalized Petersen graphs, is given., Comment: 18 pages, 0 figures

It is shown that every connected vertex-transitive graph of order $4p$, where $p$ is a prime, is hamiltonian with the exception of the Coxeter graph which is known to possess a Hamilton path., Comment: 17 pages, 9 figures

Following a problem posed by Lov\'asz in 1969, it is believed that every connected vertex-transitive graph has a Hamilton path. This is shown here to be true for cubic Cayley graphs arising from groups having a $(2,s,3)$-presentation, that is, for groups $G=\la a,b| a^2=1, b^s=1, (ab)^3=1, etc. \ra$ generated by an involution $a$ and an element $b$ of order $s\geq3$ such that their product $ab$ has order 3. More precisely, it is shown that the Cayley graph $X=Cay(G,\{a,b,b^{-1}\})$ has a Hamilton cycle when $|G|$ (and thus $s$) is congruent to 2 modulo 4, and has a long cycle missing only two vertices (and thus necessarily a Hamilton path) when $|G|$ is congruent to 0 modulo 4., Comment: 13 pages, 6 figures