On the diameter of the Kronecker product graph

Let $G_1$ and $G_2$ be two undirected nontrivial graphs. The Kronecker product of $G_1$ and $G_2$ denoted by $G_1\otimes G_2$ with vertex set $V(G_1)\times V(G_2)$, two vertices $x_1x_2$ and $y_1y_2$ are adjacent if and only if $(x_1,y_1)\in E(G_1)$ and $(x_2,y_2)\in E(G_2)$. This paper presents a formula for computing the diameter of $G_1\otimes G_2$ by means of the diameters and primitive exponents of factor graphs.Comment: 9 pages, 18 reference

Connectivity of Direct Products of Graphs

Let $\kappa(G)$ be the connectivity of $G$ and $G\times H$ the direct product of $G$ and $H$. We prove that for any graphs $G$ and $K_n$ with $n\ge 3$, $\kappa(G\times K_n)=min\{n\kappa(G),(n-1)\delta(G)\}$, which was conjectured by Guji and Vumar.Comment: 5 pages, accepted by Ars Com

Neighborhood complexes and Kronecker double coverings

The neighborhood complex $N(G)$ is a simplicial complex assigned to a graph $G$ whose connectivity gives a lower bound for the chromatic number of $G$. We show that if the Kronecker double coverings of graphs are isomorphic, then their neighborhood complexes are isomorphic. As an application, for integers $m$ and $n$ greater than 2, we construct connected graphs $G$ and $H$ such that $N(G) \cong N(H)$ but $\chi(G) = m$ and $\chi(H) = n$. We also construct a graph $KG_{n,k}'$ such that $KG_{n,k}'$ and the Kneser graph $KG_{n,k}$ are not isomorphic but their Kronecker double coverings are isomorphic.Comment: 10 pages. Some results concerning box complexes are deleted. to appear in Osaka J. Mat

A Tight Lower Bound for Counting Hamiltonian Cycles via Matrix Rank

For even $k$, the matchings connectivity matrix $\mathbf{M}_k$ encodes which pairs of perfect matchings on $k$ vertices form a single cycle. Cygan et al. (STOC 2013) showed that the rank of $\mathbf{M}_k$ over $\mathbb{Z}_2$ is $\Theta(\sqrt 2^k)$ and used this to give an $O^*((2+\sqrt{2})^{\mathsf{pw}})$ time algorithm for counting Hamiltonian cycles modulo $2$ on graphs of pathwidth $\mathsf{pw}$. The same authors complemented their algorithm by an essentially tight lower bound under the Strong Exponential Time Hypothesis (SETH). This bound crucially relied on a large permutation submatrix within $\mathbf{M}_k$, which enabled a "pattern propagation" commonly used in previous related lower bounds, as initiated by Lokshtanov et al. (SODA 2011). We present a new technique for a similar pattern propagation when only a black-box lower bound on the asymptotic rank of $\mathbf{M}_k$ is given; no stronger structural insights such as the existence of large permutation submatrices in $\mathbf{M}_k$ are needed. Given appropriate rank bounds, our technique yields lower bounds for counting Hamiltonian cycles (also modulo fixed primes $p$) parameterized by pathwidth. To apply this technique, we prove that the rank of $\mathbf{M}_k$ over the rationals is $4^k / \mathrm{poly}(k)$. We also show that the rank of $\mathbf{M}_k$ over $\mathbb{Z}_p$ is $\Omega(1.97^k)$ for any prime $p\neq 2$ and even $\Omega(2.15^k)$ for some primes. As a consequence, we obtain that Hamiltonian cycles cannot be counted in time $O^*((6-\epsilon)^{\mathsf{pw}})$ for any $\epsilon>0$ unless SETH fails. This bound is tight due to a $O^*(6^{\mathsf{pw}})$ time algorithm by Bodlaender et al. (ICALP 2013). Under SETH, we also obtain that Hamiltonian cycles cannot be counted modulo primes $p\neq 2$ in time $O^*(3.97^\mathsf{pw})$, indicating that the modulus can affect the complexity in intricate ways.Comment: improved lower bounds modulo primes, improved figures, to appear in SODA 201

Connectivity of Kronecker products by K2

Let $\kappa(G)$ be the connectivity of $G$. The Kronecker product $G_1\times G_2$ of graphs $G_1$ and $G_2$ has vertex set $V(G_1\times G_2)=V(G_1)\times V(G_2)$ and edge set $E(G_1\times G_2)=\{(u_1,v_1)(u_2,v_2):u_1u_2\in E(G_1),v_1v_2\in E(G_2)\}$. In this paper, we prove that $\kappa(G\times K_2)=\textup{min}\{2\kappa(G), \textup{min}\{|X|+2|Y|\}\}$, where the second minimum is taken over all disjoint sets $X,Y\subseteq V(G)$ satisfying (1)$G-(X\cup Y)$ has a bipartite component $C$, and (2) $G[V(C)\cup \{x\}]$ is also bipartite for each $x\in X$.Comment: 6 page