Michael Tarsi

COLORINGS AND ORIENTATIONS OF GRAPHS

Bounds for the chromatic number and for some related parameters of a graph are obtained by applying algebraic techniques. In particular, the following result is proved: IfG is a directed graph with maximum outdegreed, and if the number of Eulerian subgraphs ofG with an even number of edges differs from the number of Eulerian subgraphs with an odd number of edges then for any assignment of a setS(v) ofd+1 colors for each vertexv ofG there is a legal vertex-coloring ofG assigning to each vertexv a color fromS(v).

Group connectivity of graphs: a nonhomogeneous analogue of nowhere-zero flow properties

Abstract Let G = (V, E) be a digraph and f a mapping from E into an Abelian group A. Associated with f is its boundary ∂f, a mapping from V to A, defined by ∂f(x) = Σe leaving xf(e)−Σe entering xf(e). We say that G is A-connected if for every b: V → A with Σx ∈ Vb(x)=0 there is an f: E → A − {0} with b = ∂f. This concept is closely related to the theory of nowhere-zero flows and is being studied here in light of that theory.

Graph Decomposition is NP-Complete: A Complete Proof of Holyer's Conjecture

An H-decomposition of a graph G=(V,E) is a partition of E into subgraphs isomorphic to H. Given a fixed graph H, the H-decomposition problem is to determine whether an input graph G admits an H-decomposition. In 1980, Holyer conjectured that H-decomposition is NP-complete whenever H is connected and has three edges or more. Some partial results have been obtained since then. A complete proof of Holyers conjecture is the content of this paper. The characterization problem of all graphs H for which H-decomposition is NP-complete is hence reduced to graphs where every connected component contains at most two edges.

Structuring causal trees

Abstract Models of complex phenomena often consist of hypothetical entities called “hidden causes,” which cannot be observed directly and yet play a major role in understanding those phenomena. This paper examines the computational roles of these constructs, and addresses the question of whether they can be discovered from empirical observations. Causal models are treated as trees of binary random variables where the leaves are accessible to direct observation, and the internal nodes—representing hidden causes—account for interleaf dependencies. In probabilistic terms, every two leaves are conditionally independent given the value of some internal node between them. We show that if the mechanism which drives the visible variables is indeed tree structured, then it is possible to uncover the topology of the tree uniquely by observing pairwise dependencies among the leaves. The entire tree structure, including the strengths of all internal relationships, can be reconstructed in time proportional to n log n , where n is the number of leaves.

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.

Covering Multigraphs by Simple Circuits

Answering a question raised in [SIAM J. Comput., 10 (1981), pp. 746–750], we show that every bridgeless multigraph with v vertices and e edges can be covered by simple circuits whose total length is at most

A nowhere-zero point in linear mappings

\min ( \tfrac{5}{3} e, e + \tfrac{7}{3} v - \tfrac{7}{3} )

Flows, View Obstructions, and the Lonely Runner

. Our proof supplies an efficient algorithm for finding such a cover.

Graph decomposition is NPC - a complete proof of Holyer's conjecture

We state the following conjecture and prove it for the case whereq is a proper prime power:Let A be a nonsingular n by n matrix over the finite field GFqq≧4, then there exists a vector x in (GFq)n such that both x and Ax have no zero component.

NP-completeness of graph decomposition problems

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.