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Dragan Marusic

Publications and source records attributed to Dragan Marusic.

9 recordsLinked to original sources
arXiv2018

Hamilton cycles in vertex-transitive graphs of order a product of two primes

A step forward is made in a long standing Lovász'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 Chvátal's theorem and Jackson's theorem, to certain results on polynomial representations of quadratic residues at primitive roots in finite fields.

math.CO↗
arXiv2016

Odd extensions of transitive groups via symmetric graphs

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.

math.CO↗
arXiv2016

On cubic symmetric non-Cayley graphs with solvable automorphism groups

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.

math.CO↗
arXiv2014

Transitive Group Actions: (IM)PRIMITIVITY and Semiregular Subgroups

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.

math.GR↗
arXiv2007

On Cyclic Edge-Connectivity of Fullerenes

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 $F$.

math.CO↗
arXiv2007

On quartic half-arc-transitive metacirculants

Following Alspach and Parsons, a {\em metacirculant graph} is a graph admitting a transitive group generated by two automorphisms $ρ$ and $σ$, where $ρ$ is $(m,n)$-semiregular for some integers $m \geq 1$, $n \geq 2$, and where $σ$ normalizes $ρ$, cyclically permuting the orbits of $ρ$ in such a way that $σ^m$ 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.

math.CO↗
arXiv2007

On 2-Fold Covers of Graphs

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.

math.CO↗
arXiv2005

Hamiltonicity of Cubic Cayley Graphs

Following a problem posed by Lovász 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.

math.CO↗