Luis A. Goddyn

On (k, d)-colorings and fractional nowhere-zero flows

Let Sm denote the m-vertex simple digraph formed by m - 1 edges with a common tail. Let f(m) denote the minimum n such that every n-vertex tournament has a spanning subgraph consisting of n-m disjoint copies of Sm. We prove that m lg m - m lg lg m ≤ f(m) ≤ 4m2 - 6m for sufficiently large m.

Graphs with the circuit cover property

A circuit cover of an edge-weighted graph (G, p) is a multiset of circuits in G such that every edge e is contained in exactly p(e) circuits in the multiset. A nonnegative integer valued weight vector p is admissible if the total weight of any edge-cut is even, and no edge has more than half the total weight of any edge-cut containing it. A graph G has the circuit cover property if (G, p) has a circuit cover for every admissible weight vector p . We prove that a graph has the circuit cover property if and only if it contains no subgraph homeomorphic to Petersens graph. In particular, every 2-edge-connected graph with no subgraph homeomorphic to Petersens graph has a cycle double cover.

Packing Non-Zero A -Paths In Group-Labelled Graphs

Let G=(V,E) be an oriented graph whose edges are labelled by the elements of a group Γ and let A⊂V. An A-path is a path whose ends are both in A. The weight of a path P in G is the sum of the group values on forward oriented arcs minus the sum of the backward oriented arcs in P. (If Γ is not abelian, we sum the labels in their order along the path.) We are interested in the maximum number of vertex-disjoint A-paths each of non-zero weight. When A = V this problem is equivalent to the maximum matching problem. The general case also includes Maders S-paths problem. We prove that for any positive integer k, either there are k vertex-disjoint A-paths each of non-zero weight, or there is a set of at most 2k −2 vertices that meets each of the non-zero A-paths. This result is obtained as a consequence of an exact min-max theorem.

Flows, View Obstructions, and the Lonely Runner

We prove the following result: LetGbe an undirected graph. IfGhas a nowhere zero flow with at mostkdifferent values, then it also has one with values from the set {1, ?, k}. Whenk?5, this is a trivial consequence of Seymours “six-flow theorem”. Whenk?4 our proof is based on a lovely number theoretic problem which we call the “Lonely Runner Conjecture:” Supposekrunners having nonzero constant speeds run laps on a unit-length circular track. Then there is a time at which all runners are at least 1/(k+1) from their common starting point. This conjecture appears to have been formulated by J. Wills (Monatsch. Math.71, 1967) and independently by T. Cusick (Aequationes Math.9, 1973). This conjecture has been verified fork?4 by Cusick and Pomerance (J. Number Theory19, 1984) in a complicated argument involving exponential sums and electronic case checking. A major part of this paper is an elementary selfcontained proof of the casek=4 of the Lonely Runner Conjecture.

List edge colourings of some 1-factorable multigraphs

AbstractThe List Edge Colouring Conjecture asserts that, given any multigraphG with chromatic indexk and any set system {Se:e∈E(G)} with each |Se|=k, we can choose elementsse∈Sesuch thatse≠sfwhenevere andf are adjacent edges. Using a technique of Alon and Tarsi which involves the graph monomial

Removable Circuits in Multigraphs

\prod {\left\{ {xu - x_\upsilon :u\upsilon \in E} \right\}}

A Girth Requirement for the Double Cycle Cover Conjecture

of an oriented graph, we verify this conjecture for certain families of 1-factorable multigraphs, including 1-factorable planar graphs.

Matroids with the Circuit Cover Property

Let H be a fixed graph. We show that any H-minor free graph G of high enough girth has circular chromatic number arbitrarily close to two. Equivalently, each such graph G admits a homomorphism to a large odd circuit. In particular, graphs of high girth and of bounded genus, or of bounded tree width, are “nearly bipartite” in this sense. For example, any planar graph of girth at least 16 admits a homomorphism to a pentagon. We also obtain tight bounds on the girth of G in a few specific cases of small forbidden minors H.