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

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

Treelike snarks

V článku studujeme grafy typu snark, jejichž hrany se nedají pokrýt méně než 5 perfektními párováními. Esperet a Mazzuoccolo našli nekonečnou třídu takových grafů a zobecnili tak příklad zkonstruovaný Hägglundem. Ukážeme konstrukci jiné nekonečné třídy, získané zobecněním v odlišném směru. Důkaz, že tato třída má požadovanou vlastnost, používá prohledávání pomocí počítače. Dále ukazujeme, že grafy z této třídy (říkáme jim stromovité grafy typu snark) mají cirkulární tokové číslo $\phi_C (G)\ge5$ a mají dvojité pokrytí 5 cykly.We study snarks whose edges cannot be covered by fewer than five perfect matchings. Esperet and Mazzuoccolo found an infinite family of such snarks, generalising an example provided by Hägglund. We construct another infinite family, arising from a generalisation in a different direction. The proof that this family has the requested property is computer-assisted. In addition, we prove that the snarks from this family (we call them \emph{treelike snarks}) have circular flow number $\phi_C (G)\ge5$ and admit a 5-cycle double cover

Generalized Petersen Graphs are not PMH

In this paper, we prove that not all perfect matchings in Generalized Petersen graphs can be extended to Hamiltonian cycles