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A graph $G$ is said to be $d$-distinguishable if there is a labeling of the vertices with $d$ labels so that only the trivial automorphism preserves the labels. The smallest such $d$ is the distinguishing number, Dist($G$). A subset of vertices $S$ is a determining set for $G$ if every automorphism of $G$ is uniquely determined by its action on $S$. The size of a smallest determining set for $G$ is called the determining number, Det($G$). The orthogonality graph $\Omega_{2k}$ has vertices which are bitstrings of length $2k$ with an edge between two vertices if they differ in precisely $k$ bits. This paper shows that Det($\Omega_{2k}$) $= 2^{2k-1}$ and that if $\binom{m}{2} \geq 2k$ then $2

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. © 2011 Wiley Periodicals, Inc. J Graph Theory 69:97-113, 2012

A graph G is said to be 2 -distinguishable if there is a labeling of the vertices with two labels so that only the trivial automorphism preserves the labels. Call the minimum size of a label class in such a labeling of G the cost of 2 -distinguishing G and denote it by ρ ( G ). This paper shows that for n ≥ 5, ⌈log 2 n ⌉ + 1 ≤ ρ ( Q n ) ≤ 2⌈log 2 n ⌉ − 1, where Q n is the hypercube of dimension n .

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.

A graph $G$ is said to be $2$-distinguishable if there is a labeling of the vertices with two labels so that only the trivial automorphism preserves the label classes. The minimum size of a label class in any such labeling of $G$ is called the cost of $2$-distinguishing $G$ and is denoted by $\rho(G)$. The determining number of a graph $G$, denoted $\det(G)$, is the minimum size of a set of vertices whose pointwise stabilizer is trivial. The main result of this paper is that if $G^k$ is a $2$-distinguishable Cartesian power of a prime, connected graph $G$ on at least three vertices with $\det(G)\leq k$ and $\max\{2, \det(G)\} < \det(G^k)$, then $\rho(G^k) \in \{\det(G^k), \det(G^k)+1\}$. In particular, for $n\geq 3$, $\rho(K_3^n)\in \{ \left\lceil {\log_3 (2n+1)} \right\rceil$ $+1, \left\lceil {\log_3 (2n+1)} \right\rceil$ $+2\}$.

A geometric graph G(bar) is a simple graph drawn in the plane, on points in general position, with straight-line edges. We call G(bar) a geometric realization of the underlying abstract graph G. A geometric homomorphism 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. We say G(bar) precedes G(hat) if G(bar) and G(hat) are geometric realizations of G and there is a vertex-injective geometric homomorphism from G(bar) to G(hat). This paper develops tools to determine when two geometric realizations are comparable. Further, for 3 \leq n \leq 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 of these graphs. The paper also provides the following results for general n: the poset of P_n and C_n has a unique minimal element and a unique maximal element; if k \leq n then the poset of P_k (resp., the poset of C_k) is a subposet of the poset for P_n (resp., C_n); and the poset for K_n contains a chain of length n-2., 42 pages, 25 figures; co-author added in replaced version

This work introduces the technique of using a carefully chosen determining set to prove the existence of a distinguishing labeling using few labels. A graph $G$ is said to be $d$-distinguishable if there is a labeling of the vertex set using $1, \ldots, d$ so that no nontrivial automorphism of $G$ preserves the labels. A set of vertices $S\subseteq V(G)$ is a determining set for $G$ if every automorphism of $G$ is uniquely determined by its action on $S$. We prove that a graph is $d$-distinguishable if and only if it has a determining set that can be $(d-1)$-distinguished. We use this to prove that every Kneser graph $K_{n:k}$ with $n\geq 6$ and $k\geq 2$ is $2$-distinguishable.

A set of vertices $S$ is a determining set for a graph $G$ if every automorphism of $G$ is uniquely determined by its action on $S$. The determining number of a graph is the size of a smallest determining set. This paper describes ways of finding and verifying determining sets, gives natural lower bounds on the determining number, and shows how to use orbits to investigate determining sets. Further, determining sets of Kneser graphs are extensively studied, sharp bounds for their determining numbers are provided, and all Kneser graphs with determining number $2$, $3,$ or $4$ are given.

A graph $G$ is said to be 2-distinguishable if there is a labeling of the vertices with two labels such that only the trivial automorphism preserves the labels. Call the minimum size of a label class in such a labeling of $G$ the cost of 2-distinguishing $G$.We show that the connected, locally finite, infinite graphs with finite 2-distinguishing cost are exactly the graphs with countable automorphism group. Further we show that in such graphs the cost is less than three times the size of a smallest determining set. We also another, sharper bound on the 2-distinguishing cost, in particular for graphs of linear growth.