Paul A. Catlin

A reduction method to find spanning Eulerian subgraphs

We ask, When does a graph G have a subgraph Γ such that the vertices of odd degree in Γ form a specified set S ⊆ V(G), such that G - E(Γ) is connected? If such a subgraph can be found for a suitable choice of S, then this can be applied to problems such as finding a spanning eulerian subgraph of G. We provide a general method, with applications.

Supereulerian graphs: a survey

A graph is supereulerian if it has a spanning eulerian subgraph. There is a rduction method to determine whether a graph is supereulerian, and it can also be applied to study other concepts, e.g., hamiltonian line graphs, a certain type of double cycle cover, and the total interval number of a graph. We outline the research on supereulerian graphs, the reduction method, and its applications.

Hajós' graph-coloring conjecture: Variations and counterexamples

For each integer n ≥ 7, we exhibit graphs of chromatic number n that contain no subdivided Kn as a subgraph. However, we show that a graph with chromatic number 4 contains as a subgraph a subdivided K4 in which each triangle of the K4 is subdivided to form an odd cycle.

Hadwiger's Conjecture is True for Almost Every Graph

The contraction clique number ccl( G ) of a graph G is the maximal r for which G has a subcontraction to the complete graph K ′. We prove that for d > 2, almost every graph of order n satisfies n ((log 2 n ) 1/2 +4) -1 ⩽ ccl( G ) ⩽ n (log 2 n - d log 2 log 2 n ) 1/2 . This inequality implies the statement in the title.

Graphs without spanning closed trails

Jaeger [J. Graph Theory 3 (1979) 91-93] proved that if a graph has two edge-disjoint spanning trees, then it is supereulerian, i.e., that it has a spanning closed trail. Catlin [J. Graph Theory 12 (1988) 29-45] showed that if G is one edge short of having two edge-disjoint spanning trees, then G has a cut edge or G is supereulerian. Catlin conjectured that if a connected graph G is at most two edges short of having two edge-disjoint spanning trees, then either G is supereulerian or G can be contracted to a K2 or a K2,t for some odd integer t 1. We prove Catlin’s conjecture in a more general context. Applications to spanning trails are discussed.

A bound on the chromatic number of a graph

We give an upper bound on the chromatic number of a graph in terms of its maximum degree and the size of the largest complete subgraph. Our results extends a theorem due to Brooks.

Subgraphs of graphs, I

Let G and H be two simple graphs on p vertices. We give a sufficient condition, based on the minimum degree of the vertices of G and the maximum degree of the vertices of H, for H to be a subgraph of G.

Fractional arboricity, strength, and principal partitions in graphs and matroids

Abstract In a 1983 paper, D. Gusfield introduced a function which is called (following W.H. Cunningham, 1985) the strength of a graph or matroid. In terms of a graph G with edge set E(G) and at least one link, this is the function η(G) = minF⊆E(G) ∣F∣/(ω(G − F) − ω(G)), where the minimum is taken over all subsets F of E(G) such that ω(G − F), the number of components of G − F, is at least ω(G) + 1. In a 1986 paper, C. Payan introduced the fractional arboricity of a graph or matroid. In terms of a graph G with edge set E(G) and at least one link this function is γ(G) = maxH⊆G ∣E(H)∣/(∣V(H)∣ − ω(H)), where H runs over all subgraphs of G having at least one link. Connected graphs G for which γ(G) = η(G) were used by A. Rucinski and A. Vince in 1986 while studying random graphs. We characterize the graphs and matroids G for which γ(G) = η(G). The values of γ and η are computed for certain graphs, and a recent result of Erdos (that if each edge of G lies in a C3, then ∣E(G)∣≥ 3 2 (∣V(G)∣ − 1)) is generalized in terms of η. The principal partition of a graph was introduced in 1967 by G. Kishi and Y. Kajitani, by T. Ohtsuki, Y. Ishizaki, and H. Watanabe, and by M. Iri (all of these were published in 1968). It has been used since then for the analysis of electrical networks in which the two Kirchhoff laws and Ohms law hold, because it often allows the currents and voltage drops in the network to be completely computed with fewer measurements than are required for either of the Kirchhoff laws used alone. J. Bruno and L. Weinberg generalized the principal partition to matroids in 1971, and their generalization was refined independently by N. Tomizawa (1976) and by H. Narayanan and M.N. Vartak (1974, 1981). Here we demonstrate that γ and η are closely related to the principal partition and can be used to give a simple definition of both the principal partition and the more recent refinements of it.

Double cycle covers and the petersen graph

Let O(G) denote the set of odd-degree vertices of a graph G. Let t ϵ N and let t denote the family of graphs G whose edge set has a partition. E(g) = E1 U E2 U … U Etsuch that O(G) = O(G[Ei]) (1 ⩽ i ⩽ t). This partition is associated with a double cycle cover of G. We show that if a graph G is at most 5 edges short of being 4-edge-connected, then exactly one of these holds: G ϵ 3, G has at least one cut-edge, or G is contractible to the Petersen graph. We also improve a sufficient condition of Jaeger for G ϵ 2p+1(p ϵ N).

Supereulerian Graphs and the Petersen Graph

Any 3-edge-connected graph with at most 10 edge cuts of size 3 either has a spanning closed trail or it is contractible to the Petersen graph.