No abstract

We consider infinite graphs. The distinguishing number D(G) of a graph G is the minimum number of colours in a vertex colouring of G that is preserved only by the trivial automorphism. An analogous invariant for edge colourings is called the distinguishing index, denoted by D (G). We prove that D (G) D(G) + 1. For proper colourings, we study relevant invariants called the distinguishing chromatic number χ D (G), and the distinguishing chromatic index χ D (G), for vertex and edge colourings, respectively. We show that χ D (G) 2∆(G) − 1 for graphs with a finite maximum degree ∆(G), and we obtain substantially lower bounds for some classes of graphs with infinite motion. We also show that χ D (G) χ (G) + 1, where χ (G) is the chromatic index of G, and we prove a similar result χ D (G) χ (G) + 1 for proper total colourings. A number of conjectures are formulated.

Let R be a finite commutative ring with 1 = 0. In this article, we study the total graph of R, denoted by τ (R), determine some of its basic graph-theoretical properties, determine when it is Eulerian, and find some conditions under which this graph is isomorphic to Cay(R, Z(R)\{0}). We shall also compute the domination number of τ (R).

A vertex coloring of a graph G is called distinguishing if no non-identity automorphisms of G can preserve it. The distinguishing number of G, denoted by D(G), is the minimum number of colors required for such coloring. The distinguishing threshold of G, denoted by θ(G), is the minimum number k of colors such that every k-coloring of G is distinguishing. In this paper, we study θ(G), find its relation to the cycle structure of the automorphism group and prove that θ(G) = 2 if and only if G is isomorphic to K 2 or K 2 . Moreover, we study graphs that have the distinguishing threshold equal to 3 or more and prove that θ(G) = D(G) if and only if G is asymmetric, K n or K n . Finally, we consider Johnson scheme graphs for their distinguishing number and threshold concludes the paper.

A self-contained graph is an infinite graph which is isomorphic to one of its proper induced subgraphs. In this paper, these graphs are studied by presenting some examples and defining some of their sub-structures such as removable subgraphs and the foundation. Then, we show that the general version of graph alternative conjecture, which says every graph has infinitely many strong twins or none, can be deduced from its connected version, which says every connected graph has infinitely many connected strong twins or none. Moreover, we try to find out under what conditions on two arbitrary removable subgraphs, their union is also a removable subgraph.