Construction of cycle double covers for certain classes of graphs

We introduce two classes of graphs, Indonesian graphs and $k$-doughnut graphs. Cycle double covers are constructed for these classes. In case of doughnut graphs this is done for the values $k=1,2,3$ and 4

Reducible configurations for the cycle double cover conjecture

AbstractA CDC (cycle double cover) of a graph G is a system (C1,…,Ck) of cycles in G such that each edge of G is contained in Ci for exactly two indices i (here a cycle is a subgraph in which each vertex has an even degree). The well-known CDC conjecture states that each bridgeless graph G has a CDC. In 1985, Goddyn proved that each minimal counterexample to the CDC conjecture has girth at least 7 (later, he even obtained the lower bound 10) by showing that each circuit C of length less than 7 is reducible, i.e. if G is a graph containing C and if G′ is obtained from G by replacing C by a certain smaller subgraph, then each CDC of G′ yields a CDC of G. Here we refine Goddyn's ideas and we present some algorithms for verifying such reduction properties. By implementing these algorithms on a computer, we can prove so far that each minimal counterexample of the CDC conjecture has girth at least 12 and we can show that each minimal counterexample of the 5-CDC conjecture (each bridgeless graph has a CDC consisting of only 5 cycles) has girth at least 10. Moreover, by using a recent result of Robertson et al. (preprint), we can prove without a computer that each bridgeless cubic graph not containing the Petersen graph as a minor has a 5-CDC which can be constructed in polynomial time. This partially settles a problem of Alspach et al. (Trans. Amer. Math. Soc. 344 (1994) 131–154)

Is the five-flow conjecture almost false?

The number of nowhere zero Z_Q flows on a graph G can be shown to be a polynomial in Q, defining the flow polynomial \Phi_G(Q). According to Tutte's five-flow conjecture, \Phi_G(5) > 0 for any bridgeless G.A conjecture by Welsh that \Phi_G(Q) has no real roots for Q \in (4,\infty) was recently disproved by Haggard, Pearce and Royle. These authors conjectured the absence of roots for Q \in [5,\infty). We study the real roots of \Phi_G(Q) for a family of non-planar cubic graphs known as generalised Petersen graphs G(m,k). We show that the modified conjecture on real flow roots is also false, by exhibiting infinitely many real flow roots Q>5 within the class G(nk,k). In particular, we compute explicitly the flow polynomial of G(119,7), showing that it has real roots at Q\approx 5.0000197675 and Q\approx 5.1653424423. We moreover prove that the graph families G(6n,6) and G(7n,7) possess real flow roots that accumulate at Q=5 as n\to\infty (in the latter case from above and below); and that Q_c(7)\approx 5.2352605291 is an accumulation point of real zeros of the flow polynomials for G(7n,7) as n\to\infty.Comment: 44 pages (LaTeX2e). Includes tex file, three sty files, and a mathematica script polyG119_7.m. Many improvements from version 3, in particular Sections 3 and 4 have been mostly re-writen, and Sections 7 and 8 have been eliminated. (This material can now be found in arXiv:1303.5210.) Final version published in J. Combin. Theory

Excluded minors in cubic graphs

Let G be a cubic graph, with girth at least five, such that for every partition X,Y of its vertex set with |X|,|Y|>6 there are at least six edges between X and Y. We prove that if there is no homeomorphic embedding of the Petersen graph in G, and G is not one particular 20-vertex graph, then either G\v is planar for some vertex v, or G can be drawn with crossings in the plane, but with only two crossings, both on the infinite region. We also prove several other theorems of the same kind.Comment: 62 pages, 17 figure

Hexagonal Tilings and Locally C6 Graphs

We give a complete classification of hexagonal tilings and locally C6 graphs, by showing that each of them has a natural embedding in the torus or in the Klein bottle. We also show that locally grid graphs are minors of hexagonal tilings (and by duality of locally C6 graphs) by contraction of a perfect matching and deletion of the resulting parallel edges, in a form suitable for the study of their Tutte uniqueness.Comment: 14 figure