Emily Moore

Extending Graph Colorings

Suppose ?(G)=r and P?V(G). It is known that if the distance between any two vertices in P is at least 4, then any (r+1)-coloring of P extends to an (r+1)-coloring of all of G, but an r-coloring of P might not extend to an r-coloring of G. We show that if the distance between any two vertices in P is at least 3, then an (r+1)-coloring of P can be extended to a ?(3r+1)/2?-coloring of G. Kostochka showed that if P induces a set of k-cliques whose pairwise distance is at least 4k, then an (r+1)-coloring of P can be extended to an (r+1)-coloring of G. We give Kostochkas proof and more precise results concerning the distance required between precolored components. For example, we show that when k=r, there is a coloring extension provided the cliques have pairwise distance at least 3k. We relate the structure of the precolored components to the number of extra colors needed in a coloring extension theorem. We construct families of graphs to show that all of the above results are close to being best possible.

Extending graph colorings using no extra colors

Abstract Suppose the graph G can be r-colored using colors 1,2,…, r , so that no vertex is adjacent to two vertices colored r. If P ⊂ V ( G ) is such that the distance between any two vertices in P is at least 12, then any r-coloring of P extends to an r-coloring of all of G. If there exists an r-coloring of G in which the distance between vertices colored r is at least 4 (resp. 6), then any r-coloring of P extends to an r-coloring of all of G provided the distance between any two vertices in P is at least 8 (resp. 6). Similar results hold if P induces a set of cliques. This continues previous work of the authors (Albertson and Moore, J. Combin. Theory Ser. B 77 (1999) 83.) on extending ( r +1)-colorings of r-chromatic graphs. Examples show that the distance constraints on P are almost sharp. We show that triangle-free outerplanar graphs instantiate our results and speculate on other families of graphs that might have r-color extension theorems.

Difference sets : connecting algebra, combinatorics and geometry

Table of Contents:* Introduction * Designs * Automorphisms of designs * Introducing difference sets * Bruck-Ryser-Chowla theorem * Multipliers * Necessary group conditions * Difference sets from geometry * Families from Hadamard matrices * Representation theory * Group characters * Using algebraic number theory * Applications * Background * Notation * Hints and solutions to selected exercises * Bibliography * Index * Index of parameters

Looking for Difference Sets in Groups with Dihedral Images

AbstractWe prove four theorems about groups with a dihedral (or cyclic) image containing a difference set. For the first two, suppose G, a group of order 2p

Distance constraints in graph color extensions

\tilde q

Introducing difference sets

with p an odd prime, contains a nontrivial (v, k, λ) difference set D with order n = k − λ prime to p and self-conjugate modulo p. If G has an image of order p, then 0 ≤ 2a + ∈

Using algebraic number theory

\sqrt n