Multiple Petersen subdivisions in permutation graphs

A permutation graph is a cubic graph admitting a 1-factor M whose complement consists of two chordless cycles. Extending results of Ellingham and of Goldwasser and Zhang, we prove that if e is an edge of M such that every 4-cycle containing an edge of M contains e, then e is contained in a subdivision of the Petersen graph of a special type. In particular, if the graph is cyclically 5-edge-connected, then every edge of M is contained in such a subdivision. Our proof is based on a characterization of cographs in terms of twin vertices. We infer a linear lower bound on the number of Petersen subdivisions in a permutation graph with no 4-cycles, and give a construction showing that this lower bound is tight up to a constant factor

Gamma-polynomials of flag homology spheres

Chapter 1 contains the main definitions used in this thesis. It also includes some basic theory relating to these fundamental concepts, along with examples. Chapter 1 includes an original result, Theorem 1.5.4, answering a question of Postnikov-Reiner-Williams, which characterises the normal fans of nestohedra. Chapter 2 contains the content of the paper [2], of which Theorem 2.0.6 is the main result. As mentioned, [2] shows that the Nevo and Petersen conjecture holds for simplicial complexes in sd(Σd−1). . Chapter 3 includes the content of the paper [1], where we show that the Nevo and Petersen conjecture holds for the dual simplicial complexes to nestohedra in Theorem 3.0.4. Chapter 4 contains the content of the paper [3] in which we prove Conjecture 0.0.4 in Theorem 4.1.2 by showing that tree shifts lower the γ-polynomial of graph-associahedra. Chapter 4 also includes Theorem 4.2.1, which shows that flossing moves also lower the γ-polynomial of graph-associahedra. In Chapter 5 we include smaller results that have been made. This chapter includes a result proving Gal’s conjecture for edge subdivisions of the order complexes of Gorenstein* complexes, and shows that this result can be attributed to the work of Athanasiadis in [4]. Chapter viii INTRODUCTION 5 also includes some work we have done towards answering Question 14.3 of [26] for interval building sets

Simple realizability of complete abstract topological graphs simplified

An abstract topological graph (briefly an AT-graph) is a pair $A=(G,\mathcal{X})$ where $G=(V,E)$ is a graph and $\mathcal{X}\subseteq {E \choose 2}$ is a set of pairs of its edges. The AT-graph $A$ is simply realizable if $G$ can be drawn in the plane so that each pair of edges from $\mathcal{X}$ crosses exactly once and no other pair crosses. We show that simply realizable complete AT-graphs are characterized by a finite set of forbidden AT-subgraphs, each with at most six vertices. This implies a straightforward polynomial algorithm for testing simple realizability of complete AT-graphs, which simplifies a previous algorithm by the author. We also show an analogous result for independent $\mathbb{Z}_2$-realizability, where only the parity of the number of crossings for each pair of independent edges is specified.Comment: 26 pages, 17 figures; major revision; original Section 5 removed and will be included in another pape

Smallest snarks with oddness 4 and cyclic connectivity 4 have order 44

The family of snarks -- connected bridgeless cubic graphs that cannot be 3-edge-coloured -- is well-known as a potential source of counterexamples to several important and long-standing conjectures in graph theory. These include the cycle double cover conjecture, Tutte's 5-flow conjecture, Fulkerson's conjecture, and several others. One way of approaching these conjectures is through the study of structural properties of snarks and construction of small examples with given properties. In this paper we deal with the problem of determining the smallest order of a nontrivial snark (that is, one which is cyclically 4-edge-connected and has girth at least 5) of oddness at least 4. Using a combination of structural analysis with extensive computations we prove that the smallest order of a snark with oddness at least 4 and cyclic connectivity 4 is 44. Formerly it was known that such a snark must have at least 38 vertices [J. Combin. Theory Ser. B 103 (2013), 468--488] and one such snark on 44 vertices was constructed by Lukot'ka et al. [Electron. J. Combin. 22 (2015), #P1.51]. The proof requires determining all cyclically 4-edge-connected snarks on 36 vertices, which extends the previously compiled list of all such snarks up to 34 vertices [J. Combin. Theory Ser. B, loc. cit.]. As a by-product, we use this new list to test the validity of several conjectures where snarks can be smallest counterexamples.Comment: 21 page

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)