Suppose Γ is a group acting on a set X. A k-labeling of X is a mapping c : X → {1, 2, . . . , k}. A labeling c of X is distinguishing (with respect to the action of Γ ) if for any g ∈ Γ , g = id X , there exists an element x ∈ X such that c(x) = c(g(x)). The distinguishing number, D Γ (X), of the action of Γ on X is the minimum k for which there is a k-labeling which is distinguishing. This paper studies the distinguishing number of the linear group GL n (K) over a field K acting on the vector space K n and the distinguishing number of the automorphism group Aut(G) of a graph G acting on V (G). The latter is called the distinguishing number of the graph G and is denoted by D(G). We determine the value of D GL n (K) (K n ) for all fields K and integers n. For the distinguishing number of graphs, we study the possible value of the distinguishing number of a graph in terms of its automorphism group, its maximum degree, and other structure properties. It is proved that if Aut(G) = S n and each orbit of Aut(G) has size less than n 2 , then D(G) = n 1/k for some positive integer k. 627 is a complete graph, regular complete bipartite graph, or C 5 . We introduce the notion of uniquely distinguishable graphs and study the distinguishing number of disconnected graphs.

Abstract:A graph G = (V , E) is called (k, k )-total weight choosable if the following holds: For any total list assignment L which assigns to each vertex x a set L(x) of k real numbers, and assigns to each edge e a set L(e) of k real numbers, there is a mapping f : V ∪E → R such that f (y) ∈ L(y) for any y ∈ V ∪E and for any two adjacent vertices x, x , e∈E(x) f (e)+f (x) = e∈E(x ) f (e)+f (x ). We conjecture that every graph is (2, 2)-total weight choosable and every graph without isolated edges is (1, 3)-total weight choosable. It follows from results in [7] that complete graphs, complete bipartite graphs, trees other than K 2 are (1, 3)-total weight choosable. Also a graph G obtained from an arbitrary graph H by subdividing each edge with at least three vertices is (1, 3)-total weight choosable. This article proves that complete graphs, trees, generalized theta graphs are (2, 2)-total weight choosable. We also prove that for any graph H, a graph G obtained from H by subdividing each edge with at least two vertices is (2, 2)-total weight

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