Hal A. Kierstead

Hamiltonian chains in hypergraphs

A graph G is said to be Pt-free if it does not contain an induced path on t vertices. The i-center Ci(G) of a connected graph G is the set of vertices whose distance from any vertex in G is at most i. Denote by I(t) the set of natural numbers i, ⌊t-2⌋ ≤ i ≤ t - 2, with the property that, in every connected Pt-free graph G, the i-center Ci(G) of G induces a connected subgraph of G. In this article, the sharp upper bound on the diameter of G[Ci(G)] is established for every i ∈ I(t). The sharp lower bound on I(t) is obtained consequently.

The linearity of first-fit coloring of interval graphs

It is shown that First-Fit coloring requires at most

A simple competitive graph coloring algorithm III

40\omega

A short proof of the hajnal–szemerédi theorem on equitable colouring

colors to color an interval graph with clique size

Coloring interval graphs with First-Fit

\omega

A polynomial time approximation algorithm for Dynamic Storage Allocation

. It follows that a polynomial time approximation algorithm for Dynamic Storage Allocation due to Chrobak and Slusarek has a constant performance ratio of 80.

On k-ordered Hamiltonian graphs

We consider the following game played on a finite graph G. Let r and d be positive integers. Two players, Alice and Bob, alternately color the vertices of G, using colors from a set X, with |X| = r. A color α ∈ X is legal for an uncolored vertex v if by coloring v with α, the subgraph induced by all vertices of color α has maximum degree at most d. Each player is required to color legally on each turn. Alice wins the game if all vertices of the graph are legally colored. Bob wins if there comes a time when there exists an uncolored vertex which cannot be legally colored. We show that if G is planar, then Alice has a winning strategy for this game when r = 3 and d ≥ 132. We also show that for sufficiently large d, if G is a planar graph without a 4-cycle or with girth at least 5, then Alice has a winning strategy for the game when r = 2.

Orderings on graphs and game coloring number

A proper vertex colouring of a graph is equitable if the sizes of colour classes differ by at most one. We present a new shorter proof of the celebrated Hajnal–Szemeredi theorem: for every positive integer r, every graph with maximum degree at most r has an equitable colouring with r+1 colours. The proof yields a polynomial time algorithm for such colourings.

Explicit matchings in the middle levels of the Boolean lattice

Abstract Improved bounds on the performance of the on-line graph coloring algorithm First-Fit on interval graphs are obtained.

Fibres and ordered set coloring

Abstract We use an on-line algorithm for coloring interval graphs to construct a polynomial time approximation algorithm WIC for Dynamic Storage Allocation. The performance ratio for WIC is at most six; the best previous upper bound on the performance ratio for a polynomial time approximation algorithm for Dynamic Storage Allocation had been 80.