Debra L. Boutin

Automorphisms and Distinguishing Numbers of Geometric Cliques

A geometric automorphism is an automorphism of a geometric graph that preserves crossings and noncrossings of edges. We prove two theorems constraining the action of a geometric automorphism on the boundary of the convex hull of a geometric clique. First, any geometric automorphism that fixes the boundary of the convex hull fixes the entire clique. Second, if the boundary of the convex hull contains at least four vertices, then it is invariant under every geometric automorphism. We use these results, and the theory of determining sets, to prove that every geometric n-clique in which n≥7 and the boundary of the convex hull contains at least four vertices is 2-distinguishable.

Geometric graph homomorphisms

A geometric graph is a simple graph drawn on points in the plane, in general position, with straightline edges. A geometrichomomorphism from to is a vertex map that preserves adjacencies and crossings. This work proves some basic properties of geometric homomorphisms and defines the geochromatic number as the minimum n so that there is a geometric homomorphism from to a geometric n-clique. The geochromatic number is related to both the chromatic number and to the minimum number of plane layers of . By providing an infinite family of bipartite geometric graphs, each of which is constructed of two plane layers, which take on all possible values of geochromatic number, we show that these relationships do not determine the geochromatic number. This article also gives necessary (but not sufficient) and sufficient (but not necessary) conditions for a geometric graph to have geochromatic number at most four. As a corollary, we get precise criteria for a bipartite geometric graph to have geochromatic number at most four. This article also gives criteria for a geometric graph to be homomorphic to certain geometric realizations of K2, 2 and K3, 3.

The thickness and chromatic number of r-inflated graphs

A graph has thickness t if the edges can be decomposed into t and no fewer planar layers. We study one aspect of a generalization of Ringels famous Earth-Moon problem: what is the largest chromatic number of any thickness-2 graph? In particular, given a graph G we consider the r-inflation of G and find bounds on both the thickness and the chromatic number of the inflated graphs. In some instances, the best possible bounds on both the chromatic number and thickness are achieved. We end with several open problems.

Lower bounds for constant degree independent sets

Let ?? denote the maximum number of independent vertices all of which have the same degree. We provide lower bounds for ?? for graphs that are planar, maximal planar, of bounded degree, or trees.

Posets of geometric graphs

A geometric graph Ḡ is a simple graph drawn in the plane, on points in general position, with straight-line edges. We call Ḡ a geometric realization of the underlying abstract graph G . A geometric homomorphism f : Ḡ → H is a vertex map that preserves adjacencies and crossings (but not necessarily non-adjacencies or non-crossings). This work uses geometric homomorphisms to introduce a partial order on the set of isomorphism classes of geometric realizations of an abstract graph G . Set Ḡ ≼ Ĝ if Ḡ and Ĝ are geometric realizations of G and there is a vertex-injective geometric homomorphism f : Ḡ → Ĝ . This paper develops tools to determine when two geometric realizations are comparable. Further, for 3 ≤ n ≤ 6, this paper provides the isomorphism classes of geometric realizations of P n , C n and K n , as well as the Hasse diagrams of the geometric homomorphism posets (resp., P n , C n , K n ) of these graphs. The paper also provides the following results for general n : each of P n and C n has a unique minimal element and a unique maximal element; if k ≤ n then P k (resp., C k ) is a subposet of P n (resp., C n ); and K n contains a chain of length n − 2.

More results on r-inflated graphs: Arboricity, thickness, chromatic number and fractional chromatic number

The r -inflation of a graph G is the lexicographic product G with K r . A graph is said to have thickness t if its edges can be partitioned into t sets, each of which induces a planar graph, and t is smallest possible. In the setting of the r -inflation of planar graphs, we investigate the generalization of Ringels famous Earth-Moon problem: What is the largest chromatic number of any thickness-t graph? In particular, we study classes of planar graphs for which we can determine both the thickness and chromatic number of their 2-inflations, and provide bounds on these parameters for their r -inflations. Moreover, in the same setting, we investigate arboricity and fractional chromatic number as well. We end with a list of open questions.

The Cost of Distinguishing Graphs

In a graph a set of vertices that is stabilized setwise by only the trivial automorphism is called a distinguishing class. Not every graph has such a set, but if it does, we call its minimum size the distinguishing cost. Many families of graphs have such sets, and for some families the distinguishing costs are surprisingly small. The talk begins with a survey of results about about the distinguishing cost for finite and infinite graphs. Then it concentrates on infinite graphs with finite cost and new bounds on the distinguishing cost of graphs with linear growth, two ends and either infinite automorphism group, or finite group with infinite motion. There remain interesting, unresolved problems.