Arthur H. Busch

Packing of graphic n -tuples

An n-tuple π (not necessarily monotone) is graphic if there is a simple graph G with vertex set {v1, …, vn} in which the degree of vi is the ith entry of π. Graphic n-tuples (d, …, d) and (d, …, d) pack if there are edge-disjoint n-vertex graphs G1 and G2 such that d(vi) = d and d(vi) = d for all i. We prove that graphic n-tuples π1 and π2 pack if , where Δand δdenote the largest and smallest entries in π1 + π2 (strict inequality when δ = 1); also, the bound is sharp. Kundu and Lovasz independently proved that a graphic n-tuple π is realized by a graph with a k-factor if the n-tuple obtained by subtracting k from each entry of π is graphic; for even n we conjecture that in fact some realization has k edge-disjoint 1-factors. We prove the conjecture in the case where the largest entry of π is at most n/2 + 1 and also when k⩽3.

A characterization of triangle-free tolerance graphs

We prove that a triangle-free graph G is a tolerance graph if and only if there exists a set of consecutively ordered stars that partition the edges of G. Since tolerance graphs are weakly chordal, a tolerance graph is bipartite if and only if it is triangle-free. We, therefore, characterize those tolerance graphs that are also bipartite. We use this result to show that in general, the class of interval bigraphs properly contains tolerance graphs that are triangle-free (and hence bipartite).

A Min–Max Property of Chordal Bipartite Graphs with Applications

We show that if G is a bipartite graph with no induced cycles on exactly 6 vertices, then the minimum number of chain subgraphs of G needed to cover E(G) equals the chromatic number of the complement of the square of line graph of G. Using this, we establish that for a chordal bipartite graph G, the minimum number of chain subgraphs of G needed to cover E(G) equals the size of a largest induced matching in G, and also that a minimum chain subgraph cover can be computed in polynomial time. The problems of computing a minimum chain cover and a largest induced matching are NP-hard for general bipartite graphs. Finally, we show that our results can be used to efficiently compute a minimum chain subgraph cover when the input is an interval bigraph.

Recognizing bipartite tolerance graphs in linear time

A graph G = (V,E) is a tolerance graph if each vertex v ∈ V can be associated with an interval of the real line Iv and a positive real number tv in such a way that (uv) ∈ E if and only if |Iv ∩ Iu| ≥ min(tv, tu). No algorithm for recognizing tolerance graphs in general is known. In this paper we present an O(n + m) algorithm for recognizing tolerance graphs that are also bipartite, where n and m are the number vertices and edges of the graph, respectively. We also give a new structural characterization of these graphs based on the algorithm.

New min-max theorems for weakly chordal and dually hordal graphs

A distance-k matching in a graph G is matching M in which the distance between any two edges of M is at least k. A distance-2 matching is more commonly referred to as an induced matching. In this paper, we show that when G is weakly chordal, the size of the largest induced matching in G is equal to the minimum number of co-chordal subgraphs of G needed to cover the edges of G, and that the co-chordal subgraphs of a minimum cover can be found in polynomial time. Using similar techniques, we show that the distance-k matching problem for k > 1 is tractable for weakly chordal graphs when k is even, and is NP-hard when k is odd. For dually chordal graphs, we use properties of hypergraphs to show that the distance-k matching problem is solvable in polynomial time whenever k is odd, and NP-hard when k is even. Motivated by our use of hypergraphs, we define a class of hypergraphs which lies strictly in between the well studied classes of acyclic hypergraphs and normal hypergraphs.

Improved Sufficient Conditions for the Existence of Anti-Directed Hamiltonian Cycles in Digraphs

Let D be a directed graph of order n. An anti-directed (hamiltonian) cycle H in D is a (hamiltonian) cycle in the graph underlying D such that no pair of consecutive arcs in H form a directed path in D. In this paper we give sufficient conditions for the existence of anti-directed hamiltonian cycles. Specifically, we prove that a directed graph D of even order n with minimum indegree and outdegree greater than

A Note on the Number of Hamiltonian Paths in Strong Tournaments

{\frac{1}{2}n + 7\sqrt{n}/3}

Transitive partitions in realizations of tournament score sequences

contains an anti-directed hamiltonian cycle. In addition, we show that D contains anti-directed cycles of all possible (even) lengths when n is sufficiently large and has minimum in- and out-degree at least