Sean McGuinness

Colouring Arcwise Connected Sets in the Plane II

Abstract. Let ? be the family of finite collections ? where ? is a collection of bounded, arcwise connected sets in ℝ2 where for any S,T∈? such that S∩T≠?, it holds that S∩T is arcwise connected. Given ? is triangle-free, and provided the chromatic number χ(G) of the intersection graph G=G(?) of ? is sufficiently large, there exists α>1 independent of ? such that there is a subcollection ?′⊂? of at most 5 sets with the property that the sets of ? surrounded by ?′ induce an intersection graph H where .

Removable Circuits in Multigraphs

We prove the following conjecture of Bill Jackson (J. London Math. Soc. (2)21, 1980, 391).If G is a2-connected multigraph with minimum degree at least4and containing no Petersen minor, then G contains a circuit C such that G?E(C) is2-connected. In fact,Ghas at leasttwoedge-disjoint circuits which can serve asC. Until now, the conjecture had been verified only for planar graphs and for simple graphs.

Double covers of cubic graphs with oddness 4

We prove that a cubic 2-connected graph which has a 2-factor containing exactly 4 odd cycles has a cycle double cover.

On bounding the chromatic number of L-graphs

Abstract We show that the intersection graph of a collection of subsets of the plane, where each subset forms an “L” shape whose vertical stem is infinite, has its chromatic number χ bounded by a function of the order of its largest clique ω, where it is shown that χ⩽2 ( 14 3 )(4 ω−1 −1) . This proves a special case of a conjecture of Gyarfas and Lehel.

Gallai's Conjecture For Graphs of Girth at Least Four

In 1966, Gallai conjectured that for any simple, connected graph G having n vertices, there is a path-decomposition of G having at most paths. In this article, we show that for any simple graph G having girth , there is a path-decomposition of G having at most paths, where is the number of vertices of odd degree in G and is the number of nonisolated vertices of even degree in G.

The greedy clique decomposition of a graph

We prove that if maximal cliques are removed one by one from any graph with n vertices, then the graph will be empty after at most n2/4 steps. This proves a conjecture of Winkler.

CIRCUITS THROUGH COCIRCUITS IN A GRAPH WITH EXTENSIONS TO MATROIDS

We show that for any k-connected graph having cocircumference c*, there is a cycle which intersects every cocycle of size c*-k + 2 or greater. We use this to show that in a 2-connected graph, there is a family of at most c* cycles for which each edge of the graph belongs to at least two cycles in the family. This settles a question raised by Oxley.A certain result known for cycles and cocycles in graphs is extended to matroids. It is shown that for a k-connected regular matroid having circumference c ≥ 2k if C1 and C2 are disjoint circuits satisfying r(C1)+r(C2)=r(C1∪C2), then |C1|+|C2|≤2(c-k + 1).

Recurrent networks and a theorem of Nash-Williams

In this paper, we generalize a result of Nash-Williams concerning recurrence of locally finite networks, by extending his result to networks with possibly vertices of infinite degree.

Contractible bonds in graphs

This paper addresses a problem posed by Oxley (Matroid Theory, Cambridge University Press, Cambridge, 1992) for matroids. We shall show that if G is a 2-connected graph which is not a multiple edge, and which has no K5-minor, then G has two edge-disjoint non-trivial bonds B for which G/B is 2-connected.

Restricted greedy clique decompositions and greedy clique decompositions ofK4-free graphs

AbstractA greedy clique decomposition of a graph is obtained by removing maximal cliques from a graph one by one until the graph is empty. It has recently been shown that any greedy clique decomposition of a graph of ordern has at mostn2/4 cliques. In this paper, we extend this result by showing that for any positive integerp, 3≤p any clique decomposisitioof a graph of ordern obtained by removing maximal cliques of order at leastp one by one until none remain, in which case the remaining edges are removed one by one, has at mosttp-1(n) cliques. Heretp-1(n) is the number of edges in the Turán graph of ordern, which has no complete subgraphs of orderp.In connection with greedy clique decompositions, P. Winkler conjectured that for any greedy clique decompositionC of a graphG of ordern the sum over the number of vertices in each clique ofC is at mostn2/2. We prove this conjecture forK4-free graphs and show that in the case of equality forC andG there are only two possibilities:(i)G≃Kn/2,n/2(ii)G is complete 3-partite, where each part hasn/3 vertices. We show that in either caseC is completely determined.