ON THE GRAPHS WITH DISTINGUISHING NUMBER EQUAL LIST DISTINGUISHING NUMBER.

ALIKHANI, S.; SOLTANI, S.

The distinguishing number D(G) of a graph G is the least integer d such that G has a vertex labeling with d labels that is preserved only by the trivial automorphism. A list assignment to G is an assignment L = {L(v)} v∈V (G) of lists of labels to the vertices of G. A distinguishing L-labeling of G is a distinguishing labeling of G where the label of each vertex v comes from L(v). The list distinguishing number of G, denoted by Dl(G), is the minimum k such that every list assignment to G in which |L(v)| = k for all v ∈ V (G) yields a distinguishing L-labeling of G. In this paper, we determine the list-distinguishing number for two families of graphs. We also characterize all graphs with the distinguishing number equal the list distinguishing number. Finally, we show that this characterization works for other list numbers of a graph.

GRAPH theory; INTEGERS; NUMBER theory; GEOMETRIC vertices; AUTOMORPHISMS; GRAPH labelings

Journal of Mahani Mathematical Research Center, 2023, Vol 12, Issue 2, p411

2251-7952

Academic Journal

10.22103/jmmr.2023.20163.1333