Even Orientations and Pfaffian graphs

We give a characterization of Pfaffian graphs in terms of even orientations, extending the characterization of near bipartite non--pfaffian graphs by Fischer and Little \cite{FL}. Our graph theoretical characterization is equivalent to the one proved by Little in \cite{L73} (cf. \cite{LR}) using linear algebra arguments

A note on 2--bisections of claw--free cubic graphs

A \emph{$k$--bisection} of a bridgeless cubic graph $G$ is a $2$--colouring of its vertex set such that the colour classes have the same cardinality and all connected components in the two subgraphs induced by the colour classes have order at most $k$. Ban and Linial conjectured that {\em every bridgeless cubic graph admits a $2$--bisection except for the Petersen graph}. In this note, we prove Ban--Linial's conjecture for claw--free cubic graphs

A construction of small (q-1)-regular graphs of girth 8

In this note we construct a new infinite family of $(q-1)$-regular graphs of girth $8$ and order $2q(q-1)^2$ for all prime powers $q\ge 16$, which are the smallest known so far whenever $q-1$ is not a prime power or a prime power plus one itself.Comment: 8 pages, 2 figure

A formulation of a (q+1,8)-cage

Let $q\ge 2$ be a prime power. In this note we present a formulation for obtaining the known $(q+1,8)$-cages which has allowed us to construct small $(k,g)$--graphs for $k=q-1, q$ and $g=7,8$. Furthermore, we also obtain smaller $(q,8)$-graphs for even prime power $q$.Comment: 14 pages, 2 figure

Why emergency management should be interested in the emergence of antibiotic resistance

Bacterial epidemics and pandemics are biological risks to life every bit as significant as floods, fires, storms and earthquakes. Antibiotics have been a significant tool in the management of epidemics and pandemics (as well as for fighting general infections) since their discovery in the 1930s. Due to the development of antibiotic resistance by bacteria, we are now approaching a post-antibiotic era where our capacity to manage infectious disease, particularly bacterial epidemics and pandemics, is compromised. Despite considerable efforts by global heath organisations, we need new ways of thinking and acting on the global risk of antibiotic resistance. We argue for a rebranding of the issue to one of a disaster risk and suggest the use of the risk management process and expertise of emergency management to present a new way of thinking about this globally significant risk to life

A Characterization of Graphs with Small Palette Index

Given an edge-coloring of a graph G, we associate to every vertex v of G the set of colors appearing on the edges incident with v. The palette index of G is defined as the minimum number of such distinct sets, taken over all possible edge-colorings of G. A graph with a small palette index admits an edge-coloring which can be locally considered to be almost symmetric, since few different sets of colors appear around its vertices. Graphs with palette index 1 are r-regular graphs admitting an r-edge-coloring, while regular graphs with palette index 2 do not exist. Here, we characterize all graphs with palette index either 2 or 3 in terms of the existence of suitable decompositions in regular subgraphs. As a corollary, we obtain a complete characterization of regular graphs with palette index 3

A construction for a counterexample to the pseudo 2-factor isomorphic graph conjecture

A graph $G$ admiting a $2$-factor is \textit{pseudo $2$-factor isomorphic} if the parity of the number of cycles in all its $2$-factors is the same. In [M. Abreu, A.A. Diwan, B. Jackson, D. Labbate and J. Sheehan. Pseudo $2$-factor isomorphic regular bipartite graphs. Journal of Combinatorial Theory, Series B, 98(2) (2008), 432-444.] some of the authors of this note gave a partial characterisation of pseudo $2$-factor isomorphic bipartite cubic graphs and conjectured that $K_{3,3}$, the Heawood graph and the Pappus graph are the only essentially $4$-edge-connected ones. In [J. Goedgebeur. A counterexample to the pseudo $2$-factor isomorphic graph conjecture. Discr. Applied Math., 193 (2015), 57-60.] Jan Goedgebeur computationally found a graph $\mathscr{G}$ on $30$ vertices which is pseudo $2$-factor isomorphic cubic and bipartite, essentially $4$-edge-connected and cyclically $6$-edge-connected, thus refuting the above conjecture. In this note, we describe how such a graph can be constructed from the Heawood graph and the generalised Petersen graph $GP(8,3)$, which are the Levi graphs of the Fano $7_3$ configuration and the M\"obius-Kantor $8_3$ configuration, respectively. Such a description of $\mathscr{G}$ allows us to understand its automorphism group, which has order $144$, using both a geometrical and a graph theoretical approach simultaneously. Moreover we illustrate the uniqueness of this graph

An explicit formula for obtaining (q+1,8)(q+1,8)-cages and others small regular graphs of girth 8

Let $q$ be a prime power; $(q+1,8)$-cages have been constructed as incidence graphs of a non-degenerate quadric surface in projective 4-space $P(4, q)$. The first contribution of this paper is a construction of these graphs in an alternative way by means of an explicit formula using graphical terminology. Furthermore by removing some specific perfect dominating sets from a $(q+1,8)$-cage we derive $k$-regular graphs of girth 8 for $k= q-1$ and $k=q$, having the smallest number of vertices known so far

Adjacency Matrices of Configuration Graphs

In 1960, Hoffman and Singleton \cite{HS60} solved a celebrated equation for square matrices of order $n$, which can be written as $$(\kappa - 1) I_n + J_n - A A^{\rm T} = A$$ where $I_n$, $J_n$, and $A$ are the identity matrix, the all one matrix, and a $(0,1)$--matrix with all row and column sums equal to $\kappa$, respectively. If $A$ is an incidence matrix of some configuration $\cal C$ of type $n_\kappa$, then the left-hand side $\Theta(A):= (\kappa - 1)I_n + J_n - A A^{\rm T}$ is an adjacency matrix of the non--collinearity graph $\Gamma$ of $\cal C$. In certain situations, $\Theta(A)$ is also an incidence matrix of some $n_\kappa$ configuration, namely the neighbourhood geometry of $\Gamma$ introduced by Lef\`evre-Percsy, Percsy, and Leemans \cite{LPPL}. The matrix operator $\Theta$ can be reiterated and we pose the problem of solving the generalised Hoffman--Singleton equation $\Theta^m(A)=A$. In particular, we classify all $(0,1)$--matrices $M$ with all row and column sums equal to $\kappa$, for $\kappa = 3,4$, which are solutions of this equation. As a by--product, we obtain characterisations for incidence matrices of the configuration $10_3F$ in Kantor's list \cite{Kantor} and the $17_4$ configuration #1971 in Betten and Betten's list \cite{BB99}