On a family of quartic graphs: Hamiltonicity, matchings and isomorphism with circulants

A pairing of a graph $G$ is a perfect matching of the underlying complete graph $K_{G}$. A graph $G$ has the PH-property if for each one of its pairings, there exists a perfect matching of $G$ such that the union of the two gives rise to a Hamiltonian cycle of $K_G$. In 2015, Alahmadi et al. proved that the only three cubic graphs having the PH-property are the complete graph $K_{4}$, the complete bipartite graph $K_{3,3}$, and the $3$-dimensional cube $\mathcal{Q}_{3}$. Most naturally, the next step is to characterise the quartic graphs that have the PH-property, and the same authors mention that there exists an infinite family of quartic graphs (which are also circulant graphs) having the PH-property. In this work we propose a class of quartic graphs on two parameters, $n$ and $k$, which we call the class of accordion graphs $A[n,k]$, and show that the quartic graphs having the PH-property mentioned by Alahmadi et al. are in fact members of this general class of accordion graphs. We also study the PH-property of this class of accordion graphs, at times considering the pairings of $G$ which are also perfect matchings of $G$. Furthermore, there is a close relationship between accordion graphs and the Cartesian product of two cycles. Motivated by a recent work by Bogdanowicz (2015), we give a complete characterisation of those accordion graphs that are circulant graphs. In fact, we show that $A[n,k]$ is not circulant if and only if both $n$ and $k$ are even, such that $k\geq 4$.Comment: 17 pages, 9 figure

A note on fractional covers of a graph

A fractional colouring of a graph $G$ is a function that assigns a non-negative real value to all possible colour-classes of $G$ containing any vertex of $G$, such that the sum of these values is at least one for each vertex. The fractional chromatic number is the minimum sum of the values assigned by a fractional colouring over all possible such colourings of $G$. Introduced by Bosica and Tardif, fractional covers are an extension of fractional colourings whereby the real-valued function acts on all possible subgraphs of $G$ belonging to a given class of graphs. The fractional chromatic number turns out to be a special instance of the fractional cover number. In this work we investigate fractional covers acting on $(k+1)$-clique-free subgraphs of $G$ which, although sharing some similarities with fractional covers acting on $k$-colourable subgraphs of $G$, they exhibit some peculiarities. We first show that if a simple graph $G_2$ is a homomorphic image of a simple graph $G_1$, then the fractional cover number defined on the $(k+1)$-clique-free subgraphs of $G_1$ is bounded above by the corresponding number of $G_2$. We make use of this result to obtain bounds for the associated fractional cover number of graphs that are either $n$-colourable or $a\!:\!b$-colourable.Comment: 8 page

The Erd\H{o}s--Faber--Lov\'{a}sz Conjecture revisited

The Erd\H{o}s--Faber--Lov\'{a}sz Conjecture, posed in 1972, states that if a graph $G$ is the union of $n$ cliques of order $n$ (referred to as defining $n$-cliques) such that two cliques can share at most one vertex, then the vertices of $G$ can be properly coloured using $n$ colours. Although still open after almost 50 years, it can be easily shown that the conjecture is true when every shared vertex belongs to exactly two defining $n$-cliques. We here provide a quick and easy algorithm to colour the vertices of $G$ in this case, and discuss connections with clique-decompositions and edge-colourings of graphs.Comment: 6 page

The super-connectivity of Johnson graphs

For positive integers $n,k$ and $t$, the uniform subset graph $G(n, k, t)$ has all $k$-subsets of $\{1,2,\ldots, n\}$ as vertices and two $k$-subsets are joined by an edge if they intersect at exactly $t$ elements. The Johnson graph $J(n,k)$ corresponds to $G(n,k,k-1)$, that is, two vertices of $J(n,k)$ are adjacent if the intersection of the corresponding $k$-subsets has size $k-1$. A super vertex-cut of a connected graph is a set of vertices whose removal disconnects the graph without isolating a vertex and the super-connectivity is the size of a minimum super vertex-cut. In this work, we fully determine the super-connectivity of the family of Johnson graphs $J(n,k)$ for $n\geq k\geq 1$

On the inverse of the adjacency matrix of a graph

A real symmetric matrix G with zero diagonal encodes the adjacencies of the vertices of a graph G with weighted edges and no loops. A graph associated with a n × n non–singular matrix with zero entries on the diagonal such that all its (n − 1) × (n − 1) principal submatrices are singular is said to be a NSSD. We show that the class of NSSDs is closed under taking the inverse of G. We present results on the nullities of one– and two–vertex deleted subgraphs of a NSSD. It is shown that a necessary and sufficient condition for two–vertex deleted subgraphs of G and of the graph (G−1) associated with G−1 to remain NSSDs is that the submatrices belonging to them, derived from G and G−1, are inverses. Moreover, an algorithm yielding what we term plain NSSDs is presented. This algorithm can be used to determine if a graph G with a terminal vertex is not a NSSDpeer-reviewe

The adjacency matrices of complete and nutful graphs

A real symmetric matrix G with zero entries on its diagonal is an adjacency matrix associated with a graph G (with weighted edges and no loops) if and only if the non-zero entries correspond to edges of G. An adjacency matrix G belongs to a generalized-nut graph G if every entry in a vector in the nullspace of G is non-zero. A graph G is termed NSSD if it corresponds to a non-singular adjacency matrix G with a singular deck {G- v}, where G- v is the submatrix obtained from G by deleting the vth row and column. An NSSD G whose deck consists of generalized- nut graphs with respect to G is referred to as a G-nutful graph. We prove that a G-nutful NSSD is equivalent to having a NSSD with G-1 as the adjacency matrix of the complete graph. If the entries of G for a G-nutful graph are restricted to 0 or 1, then the graph is known as nuciferous, a concept that has arisen in the context of the quantum mechanical theory of the conductivity of non-singular Carbon molecules according to the SSP model. We characterize nuciferous graphs by their inverse and the nullities of their one- and two-vertex deleted subgraphs. We show that a G-nutful graph is a NSSD which is either K2 or has no pendant edges. Moreover, we reconstruct a labelled NSSD either from the nullspace generators of the ordered one-vertex deleted subgraphs or from the determinants of the ordered two-vertex deleted subgraphs.peer-reviewe

Complete graphs with zero diagonal inverse

If the inverse of a non-singular real symmetric matrix that represents an edge-weighted graph with no loops has zero diagonal, then the inverse itself is the matrix of a loopless graph. Here we show that such graphs having non-zero weight on each edge always exist if their number of vertices is at least 6.peer-reviewe