arXiv subjects

Simon Spacapan

Publications and source records attributed to Simon Spacapan.

6 recordsLinked to original sources

arXiv2026

Hamiltonicity of Cartesian products of trees with odd paths

A $\{P_2,P_3\}$-factor in a graph $G$ is a factor of $G$ in which every component is a path on two or three vertices. Let $T\Box P_n$ be the Cartesian product of a tree $T$ and a path on $n$ vertices. Kao and Weng proved that $T\Box P_n$ is hamiltonian if $T$ has a path factor and $n$ is a sufficiently large even integer. In this article we prove that, for every odd $n$, there exists a tree $T$ of maximum degree 4 that has a $\{P_2,P_3\}$-factor such that $T\Box P_n$ is not hamiltonian, thereby refuting a conjecture by Kao and Weng.

math.CO↗

arXiv2019

A counterexample to prism-hamiltonicity of 3-connected planar graphs

The prism over a graph $G$ is the Cartesian product of $G$ with the complete graph $K_2$. A graph $G$ is hamiltonian if there exists a spanning cycle in $G$, and $G$ is prism-hamiltonian if the prism over $G$ is hamiltonian. In [M.~Rosenfeld, D.~Barnette, Hamiltonian circuits in certain prisms, Discrete Math. 5 (1973), 389--394] the authors conjectured that every 3-connected planar graph is prism-hamiltonian. We construct a counterexample to the conjecture.

math.CO↗

arXiv2018

The domination number of plane triangulations

We introduce a class of plane graphs called weak near-triangulations, and prove that this class is closed under certain graph operations. Then we use the properties of weak near-triangulations to prove that every plane triangulation on $n>6$ vertices has a dominating set of size at most $17n/53$. This improves the bound $n/3$ obtained by Matheson and Tarjan.

math.CO↗

arXiv2015

On Disjoint hypercubes in Fibonacci cubes

The {\em Fibonacci cube} of dimension $n$, denoted as $\Gamma\_n$, is the subgraph of $n$-cube $Q\_n$ induced by vertices with no consecutive 1's. We study the maximum number of disjoint subgraphs in $\Gamma\_n$ isomorphic to $Q\_k$, and denote this number by $q\_k(n)$. We prove several recursive results for $q\_k(n)$, in particular we prove that $q\_{k}(n) = q\_{k-1}(n-2) + q\_{k}(n-3)$. We also prove a closed formula in which $q\_k(n)$ is given in terms of Fibonacci numbers, and finally we give the generating function for the sequence $\{q\_{k}(n)\}\_{n=0}^{ \infty}$.

math.CO↗

arXiv2012

A characterization of the edge connectivity of direct products of graphs

The direct product of graphs $G=(V(G),E(G))$ and $H=(V(H),E(H))$ is the graph, denoted as $G\times H$, with vertex set $V(G\times H)=V(G)\times V(H)$, where vertices $(x_1,y_1)$ and $(x_2,y_2)$ are adjacent in $G\times H$ if $x_1x_2\in E(G)$ and $y_1y_2\in E(H)$. The edge connectivity of a graph $G$, denoted as $\lambda(G)$, is the size of a minimum edge-cut in $G$. We introduce a function $\psi$ and prove the following formula %for the edge-connectivity of direct products $$\lambda (G\times H)=\min {2\lambda(G)|E(H)|,2\lambda(H)|E(G)|,\delta(G\times H), \psi(G,H), \psi(H,G)} .$$ We also describe the structure of every minimum edge-cut in $G\times H$.

math.CO↗

arXiv2008

Degenerate and star colorings of graphs on surfaces

We study the degenerate, the star and the degenerate star chromatic numbers and their relation to the genus of graphs. As a tool we prove the following strengthening of a result of Fertin et al.: If $G$ is a graph of maximum degree $Δ$, then $G$ admits a degenerate star coloring using $O(Δ^{3/2})$ colors. We use this result to prove that every graph of genus $g$ admits a degenerate star coloring with $O(g^{3/5})$ colors. It is also shown that these results are sharp up to a logarithmic factor.

math.CO↗