Lower bound for the size of maximal nontraceable graphs

Let g(n) denote the minimum number of edges of a maximal nontraceable graph of order n. Dudek, Katona and Wojda (2003) showed that g(n)\geq\ceil{(3n-2)/2}-2 for n\geq 20 and g(n)\leq\ceil{(3n-2)/2} for n\geq 54 as well as for n\in I={22,23,30,31,38,39, 40,41,42,43,46,47,48,49,50,51}. We show that g(n)=\ceil{(3n-2)/2} for n\geq 54 as well as for n\in I\cup{12,13} and we determine g(n) for n\leq 9.Comment: 10 pages, 3 figure

Nontraceable detour graphs

AbstractThe detour order (of a vertex v) of a graph G is the order of a longest path (beginning at v). The detour sequence of G is a sequence consisting of the detour orders of its vertices. A graph is called a detour graph if its detour sequence is constant. The detour deficiency of a graph G is the difference between the order of G and its detour order. Homogeneously traceable graphs are therefore detour graphs with detour deficiency zero. In this paper, we give a number of constructions for detour graphs of all orders greater than 17 and all detour deficiencies greater than zero. These constructions are used to give examples of nontraceable detour graphs with chromatic number k, k⩾2, and girths up to 7. Moreover we show that, for all positive integers l⩾1 and k⩾3, there are nontraceable detour graphs with chromatic number k and detour deficiency l

Nested locally Hamiltonian graphs and the Oberly-Sumner conjecture

Please read abstract in the article.The DST-NRF Centre of Excellence in Mathematical and Statistical Sciences (CoE-MaSS) and the National Research Foundation of S.A.https://www.dmgt.uz.zgora.plam2023Mathematics and Applied Mathematic

Hamiltonicity of locally hamiltonian and locally traceable graphs

Please read abstract in the article.The University of South Africa and the National Research Foundation of South Africa for their sponsorship of the Salt Rock Workshops of 28 July–10 August 2013 and 20–30 January 2016, which contributed towards results in this paper. The authors thank the DST-NRF Centre of Excellence in Mathematical and Statistical Sciences (CoE-MaSS) for financial support, grant number BA2017/268. Opinions expressed and conclusions arrived at are those of the authors and are not necessarily to be attributed to the CoE-MaSS. This material is based upon the third author’s work supported by the National Research Foundation of S.A. under Grant number 81075 and the second author’s work supported by the National Research Foundation of S.A. under Grant number 107668.http://www.elsevier.com/locate/dam2019-02-19hj2018Mathematics and Applied Mathematic

On the strong path partition conjecture

The detour order of a graph G, denoted by (G), is the order of a longest path in G. If a and b are positive integers and the vertex set of G can be partitioned into two subsets A and B such that (hAi) ≤ a and (hBi) ≤ b, we say that (A,B) is an (a, b)-partition of G. If equality holds in both instances, we call (A,B) an exact (a, b)-partition. The Path Partition Conjecture (PPC) asserts that if G is any graph and a, b any pair of positive integers such that (G) = a + b, then G has an (a, b)-partition. The Strong PPC asserts that under the same circumstances G has an exact (a, b)-partition. While a substantial body of work in support of the PPC has been developed over the past three decades, no results on the Strong PPC have yet appeared in the literature. In this paper we prove that the Strong PPC holds for a ≤ 8.NSERC Discovery Grant CANADA.https://www.dmgt.uz.zgora.plam2023Mathematics and Applied Mathematic

Planar hypohamiltonian oriented graphs

In 1978 Thomassen asked whether planar hypohamiltonian oriented graphs exist. Infinite families of such graphs have since been described but for infinitely many it remained an open question whether planar hypohamiltonian oriented graphs of order exist. In this paper we develop new methods for constructing hypohamiltonian digraphs, which, combined with efficient graph generation algorithms, enable us to fully characterise the orders for which planar hypohamiltonian oriented graphs exist. Our novel methods also led us to discover the planar hypohamiltonian oriented graph of smallest order and size, as well as infinitely many hypohamiltonian orientations of maximal planar graphs. Furthermore, we answer a question related to a problem of Schiermeyer on vertex degrees in hypohamiltonian oriented graphs, and characterise all the orders for which planar hypotraceable oriented graphs exist.Research Foundation Flanders; VSC(Flemish Supercomputer Center);DST‐NRF Centre of Excellence in Mathematical and Statistical Sciences.http://wileyonlinelibrary.com/journal/jgthj2023Mathematics and Applied Mathematic

Generalised colourings of graphs

Ph.D.Please refer to full text to view abstrac