On hypohamiltonian snarks and a theorem of Fiorini

In 2003, Cavicchioli et al. corrected an omission in the statement and proof of Fiorini's theorem from 1983 on hypohamiltonian snarks. However, their version of this theorem contains an unattainable condition for certain cases. We discuss and extend the results of Fiorini and Cavicchioli et al. and present a version of this theorem which is more general in several ways. Using Fiorini's erroneous result, Steffen had shown that hypohamiltonian snarks exist for some orders n >= 10 and each even n >= 92. We rectify Steffen's proof by providing a correct demonstration of a technical lemma on flower snarks, which might be of separate interest. We then strengthen Steffen's theorem to the strongest possible form by determining all orders for which hypohamiltonian snarks exist. This also strengthens a result of Macajova and Skoviera. Finally, we verify a conjecture of Steffen on hypohamiltonian snarks up to 36 vertices

Perfect packings with complete graphs minus an edge

Let K_r^- denote the graph obtained from K_r by deleting one edge. We show that for every integer r\ge 4 there exists an integer n_0=n_0(r) such that every graph G whose order n\ge n_0 is divisible by r and whose minimum degree is at least (1-1/chi_{cr}(K_r^-))n contains a perfect K_r^- packing, i.e. a collection of disjoint copies of K_r^- which covers all vertices of G. Here chi_{cr}(K_r^-)=r(r-2)/(r-1) is the critical chromatic number of K_r^-. The bound on the minimum degree is best possible and confirms a conjecture of Kawarabayashi for large n

On almost hypohamiltonian graphs

A graph $G$ is almost hypohamiltonian (a.h.) if $G$ is non-hamiltonian, there exists a vertex $w$ in $G$ such that $G - w$ is non-hamiltonian, and $G - v$ is hamiltonian for every vertex $v \ne w$ in $G$. The second author asked in [J. Graph Theory 79 (2015) 63--81] for all orders for which a.h. graphs exist. Here we solve this problem. To this end, we present a specialised algorithm which generates complete sets of a.h. graphs for various orders. Furthermore, we show that the smallest cubic a.h. graphs have order 26. We provide a lower bound for the order of the smallest planar a.h. graph and improve the upper bound for the order of the smallest planar a.h. graph containing a cubic vertex. We also determine the smallest planar a.h. graphs of girth 5, both in the general and cubic case. Finally, we extend a result of Steffen on snarks and improve two bounds on longest paths and longest cycles in polyhedral graphs due to Jooyandeh, McKay, {\"O}sterg{\aa}rd, Pettersson, and the second author.Comment: 18 pages. arXiv admin note: text overlap with arXiv:1602.0717

The uniform face ideals of a simplicial complex

We define the uniform face ideal of a simplicial complex with respect to an ordered proper vertex colouring of the complex. This ideal is a monomial ideal which is generally not squarefree. We show that such a monomial ideal has a linear resolution, as do all of its powers, if and only if the colouring satisfies a certain nesting property. In the case when the colouring is nested, we give a minimal cellular resolution supported on a cubical complex. From this, we give the graded Betti numbers in terms of the face-vector of the underlying simplicial complex. Moreover, we explicitly describe the Boij-S\"oderberg decompositions of both the ideal and its quotient. We also give explicit formul\ae\ for the codimension, Krull dimension, multiplicity, projective dimension, depth, and regularity. Further still, we describe the associated primes, and we show that they are persistent.Comment: 34 pages, 8 figure