Planar graphs, regular graphs, bipartite graphs and Hamiltonicity

by Derek Holton and Robert E. L. Aldred Department of Mathematics and Statistics University of Otago This paper seeks to review some ideas and results relating to Hamiltonian graphs. We list the well known results which are to be found in most undergraduate graph theory courses and then consider some old theorems which are fundamental to planar graphs. By restricting our attention to 3-connected cubic planar graphs (a class of graphs of interest to Four Colour Theorists), we are able to report on recent results regarding the smallest nonHamiltonian graphs. We then consider regular graphs generally and what might be said about when the number of Hamiltonian cycles is greater than one. Another interesting class of graphs are the bipartite graphs. In general these are not Hamiltonian but there is a famous conjecture due to Barnette that suggests that 3-connected cubic bipartite planar graphs are Hamiltonian. In our final two sections we consider this along with another open conjecture due to Barnette. 1 . Introduction Although Hamiltonian graphs were named after Hamilton, this is not because of precedence. Kirkman wrote a paper in 1856 [23] in which he considered "a circuit which passes through each vertex once and only once." This paper was actually presented to the Royal Society in 1855 (see Biggs, Lloyd and Wilson [5]), while Hamilton's first communication on the subject was his letter of 1856 on the icosian calculus. Whoever originated the idea, many people have now been worrying for well over a century about graphs which have a cycle which goes through every vertex once and only once. Such graphs we'll call Hamiltonian graphs and such cycles are Hamiltonian cycles. We consider two examples, see Figure 1. These are the dodecahedron and the Petersen graph, P. In Figure lea) we show a Hamiltonian cycle by labelling the vertices in order from 1 to 20. Vertex 20 is joined to vertex 1 to complete the cycle. We've used the dodecahedron here because it was on this graph that Hamilton played cycle games. In fact he even marketed that game, though, it seems, without a great deal of financial success. On the other hand, it is well known that the Petersen graph is not Hamiltonian. A proof of this can be found by realizing that an even number of the edges 16, 2t, 39, 48 and 57 must Australasian Journal of Combinatorics 20(1999), pp.111-131 be used in a Hamiltonian cycle. This together with the property exemplified in Figure 2 quickly leads to a contradiction.