The list distinguishing number equals the distinguishing number for interval graphs

Abstract

A distinguishing coloring of a graph G is a coloring of the vertices so that every nontrivial automorphism of G maps some vertex to a vertex with a different color. The distinguishing number of G is the minimum k such that G has a distinguishing coloring where each vertex is assigned a color from {1, . . . , k}. A list assignment to G is an assignment L = {L(v)} v∈V (G) of lists of colors to the vertices of G. A distinguishing L-coloring of G is a distinguishing coloring of G where the color of each vertex v comes from L(v). The list distinguishing number of 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-coloring of G. We prove that if G is an interval graph, then its distinguishing number and list distinguishing number are equal.

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“…The introduction of the distinguishing number in 1996 by Albertson and Collins [1] was a great success; by now about one hundred papers were written motivated by this seminal paper! The core of the research has been done on the invariant D itself, either on finite [6,11,14] or infinite graphs [9,16,18]; see also the references therein. Extensions to group theory (cf.…”

The cost number and the determining number of a graph

“…The introduction of the distinguishing number was a great success; by now about one hundred papers were written motivated by this seminal paper! The core of the research has been done on the invariant D itself, either on finite [6,9,11] or infinite graphs [7,12,13]; see also the references therein.…”

Characterization of graphs with distinguishing number equal list distinguishing number

“…The introduction of the distinguishing number was a great success; by now about one hundred papers were written motivated by this seminal paper! The core of the research has been done on the invariant D itself, either on finite [3,11,12] or infinite graphs [7,13,14]; see also the references therein.…”