We prove that for every k > 0 there is an integer n0(k) such that, for every n > n0, there exists a hypohamiltonian graph which has order n and crossing number k.

Abstract: It is well-known that the largest cycles of a graph may have empty intersection. This is the case, for example, for any hypohamiltonian graph. In the literature, several important classes of graphs have been shown to contain examples with the above property. This paper investigates a (nontrivial) class of graphs which, on the contrary, admits no such example. c © 1997 John Wiley & Son...

We describe a sufficient condition for graphs used in a construction of Thomassen (which yields hypohamiltonian graphs) to produce maximally non-hamiltonian (MNH) graphs as well. Then we show that the Coxeter graph fulfils this sufficient condition, and thus applying the Thomassen’s construction to multiple copies of the Coxeter graph yields infinitely many MNH graphs with girth 7. So far, the ...

In this paper, we introduce a technique to find the cyclic path covering number of Euler graphs, by using the t-hypohamiltonian graphs. Also a special type of Euler graph in the name n-gon is also being introduced and a general result of cpcn of n-gon is also found. Finally an algorithm to find the cyclic path covering number of any Euler graph is being developed. AMS subject classification: 54...

Berge Conjecture states that every bridgeless cubic graph has 5 perfect matchings such that each edge is contained in at least one of them. In this paper, we show that Berge Conjecture holds for two classes of cubic graphs, cubic graphs with a circuit missing only one vertex and bridgeless cubic graphs with a 2-factor consisting of two circuits. The first part of this result implies that Berge ...

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 erroneou...

‎The excessive index of a bridgeless cubic graph $G$ is the least integer $k$‎, ‎such that $G$ can be covered by $k$ perfect matchings‎. ‎An equivalent form of Fulkerson conjecture (due to Berge) is that every bridgeless‎ ‎cubic graph has excessive index at most five‎. ‎Clearly‎, ‎Petersen graph is a cyclically 4-edge-connected snark with excessive index at least 5‎, ‎so Fouquet and Vanherpe as...

In 1978 Thomassen asked whether planar hypohamiltonian oriented graphs exist. Infinite families of such have since been described but for infinitely many n it remained an open question order this paper we develop new methods constructing digraphs, which, combined with efficient graph generation algorithms, enable us to fully characterise the orders which Our novel also led discover smallest and...