Even circuits in planar graphs

Projective-planar graphs with even duals

Minor-Minimal Planar Graphs of Even Branch-Width

Let k ≥ 1 be an integer, let H be a minor-minimal graph in the projective plane such that every homotopically non-trivial closed curve intersects H at least k times, and let G be the planar double cover of H obtained by lifting G into the universal covering space of the projective plane, the sphere. We prove that G is minor-minimal of branch-width 2k. We also exhibit examples of minor-minimal p...

The Alexander Polynomial of Planar Even Valence Graphs

Abstract. We show how the Alexander/Conway link polynomial occurs in the context of planar even valence graphs, refining the notion of the number of their spanning trees. Then we apply knot theory to deduce several statements about this graph polynomial, in particular estimates for its coefficients and relations between congruences of the number of vertices and number of spanning trees of the g...

Chords of longest circuits in locally planar graphs

It was conjectured by Thomassen ([B. Alspach, C. Godsil, Cycle in graphs, Ann. Discrete Math. 27 (1985)], p. 466) that every longest circuit of a 3-connected graph must have a chord. This conjecture is verified for locally 4-connected planar graphs, that is, let N be the set of natural numbers; then there is a function h : N → N such that, for every 4-connected graph G embedded in a surface S w...

Facial Circuits of Planar Graphs and Context-Free Languages

Abstrac t . It is known that a language is context-free iff it is the set of borders of the trees of recognizable set, where the border of a (labelled) tree is the word consisting of its leaf labels read from left to right. We give a generalization of this result in terms of planar graphs of bounded tree-width. Here the border of a planar graph is the word of edge labels of a path which borders...