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Dragan Marušič

Publications and source records attributed to Dragan Marušič.

8 recordsLinked to original sources
arXiv2026

Normality Of Quartic Cayley Graphs On Regular p-Groups: A CFSG-Free Approach

Relying on the Classification of Finite Simple Groups it was shown by Feng and Xu (Discrete Math., 2005) that every quartic Cayley graph of a regular $p$-group, $p \neq 2,5$, is normal. In this paper a CFSG-free proof of Feng-Xu theorem is given. Along the way it is also proved that for an arbitrary $p$-group $G$ with a minimum set $\{a,b\}$ of two generators, in the corresponding Cayley graph $\mathrm{Cay}(G,\{a,a^{-1},b,b^{-1}\})$ the induced action of vertex stabilizer on the neighbors' set is contained in the dihedral group $D_8$.

math.CO↗
arXiv2025

On 3-isoregularity of multicirculants

A graph is said to be $k$-{\em isoregular} if any two vertex subsets of cardinality at most $k$, that induce subgraphs of the same isomorphism type, have the same number of neighbors. It is shown that no $3$-isoregular bicirculant (and more generally, no locally $3$-isoregular bicirculant) of order twice an odd number exists. Further, partial results for bicirculants of order twice an even number as well as tricirculants of specific orders, are also obtained. Since $3$-isoregular graphs are necessarily strongly regular, the above result about bicirculants, among other, brings us a step closer to obtaining a direct proof of a classical consequence of the Classification of Finite Simple Groups that no simply primitive group of degree twice a prime exists for primes greater than $5$.

math.CO↗
arXiv2023

Infinite families of vertex-transitive graphs with prescribed Hamilton compression

Given a graph $X$ with a Hamilton cycle $C$, the {\em compression factor $κ(X,C)$ of $C$} is the order of the largest cyclic subgroup of $\operatorname{Aut}(C)\cap\operatorname{Aut}(X)$, and the {\em Hamilton compression $κ(X)$ of $X$ } is the maximum of $κ(X,C)$ 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 Mütze 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.

math.CO↗
arXiv2022

Intersection density of transitive groups with cyclic point stabilizers

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 $ρ(G)$ 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 Erdős-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.

math.CO↗
arXiv2021

On maximum intersecting sets in direct and wreath product of groups

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 $ρ(G)$ 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 Erdős-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 $Γ_G$, 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.

math.CO↗
arXiv2021

On intersection density of transitive groups of degree a product of two odd primes

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 $ρ(G)$ 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 $p$.

math.CO↗
arXiv2021

Intersection density of transitive groups of certain degrees

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 $ρ(G)$ 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 $1$.

math.CO↗