Irreducible pseudo 2-factor isomorphic cubic bipartite graphs

A bipartite graph is {\em pseudo 2--factor isomorphic} if all its 2--factors have the same parity of number of circuits. In \cite{ADJLS} we proved that the only essentially 4--edge-connected pseudo 2--factor isomorphic cubic bipartite graph of girth 4 is $K_{3,3}$, and conjectured \cite[Conjecture 3.6]{ADJLS} that the only essentially 4--edge-connected cubic bipartite graphs are $K_{3,3}$, the Heawood graph and the Pappus graph. There exists a characterization of symmetric configurations $n_3$ %{\bf decide notation and how to use it in the rest of the paper} due to Martinetti (1886) in which all symmetric configurations $n_3$ can be obtained from an infinite set of so called {\em irreducible} configurations \cite{VM}. The list of irreducible configurations has been completed by Boben \cite{B} in terms of their {\em irreducible Levi graphs}. In this paper we characterize irreducible pseudo 2--factor isomorphic cubic bipartite graphs proving that the only pseudo 2--factor isomorphic irreducible Levi graphs are the Heawood and Pappus graphs. Moreover, the obtained characterization allows us to partially prove the above Conjecture

An alternate description of a (q + 1; 8)-cage

Let q >= 2 be a prime power. In this note we present an alternate description of the known (q + 1; 8)-cages which has allowed us to construct small (k; g)–graphs for k =q, q-1, and g = 7; 8 in other papers on this same topic.Peer ReviewedPostprint (published version

Extending perfect matchings to Hamiltonian cycles in line graphs

A graph admitting a perfect matching has the Perfect-Matching-Hamiltonian property (for short the PMH-property) if each of its perfect matchings can be extended to a Hamiltonian cycle. In this paper we establish some sufficient conditions for a graph $G$ in order to guarantee that its line graph $L(G)$ has the PMH-property. In particular, we prove that this happens when $G$ is (i) a Hamiltonian graph with maximum degree at most $3$, (ii) a complete graph, or (iii) an arbitrarily traceable graph. Further related questions and open problems are proposed along the paper.Comment: 12 pages, 4 figure

Characterizing Minimally 1-factorable r-Regular Bipartite Graphs

AbstractThe possibilities for circuits of length 4 to appear together in a cubic bigraph are classified. That has consequences on the structure of minimally 1-factorable regular bigraphs, i.e. those in which each 1-factor lies in precisely one 1-factorization. We characterize minimally 1-factorable cubic bigraphs of girth 4

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