On the Complexity of Unique List Colourability And Abe Fixing Number Of Graphs

Abstract

Let $G$ be a finite simple $\chi$-chromatic graph and $\mathcal{L}=\{L_u{\}}_{u\in V(G)}$ be a list assignment to its vertices with $L_u\subseteq \{1,\dots ,\chi \}$. A list colouring problem $(G,\mathcal{L})$ with a unique solution for which the sum $\sum _{u\in V(G)}|L_u|$ is maximized, is called a $maximum\chi -listassignment$ of $G$. In this paper, we prove a $CircuitSimulation$ Lemma that, strictly speaking, makes it possible to simulate any Boolean function by $effective$ 3-colourings of a graph that is $polynomial-timeconstructable$ from the Boolean function itself. We use the lemma to simply prove some old results as corollaries, and also we prove that the following decision problem, related to the computation of the fixing number of a graph [Daneshgar $1997$, Daneshgar and Naserasr, Ars Combin. $69$ $(2003)$], is $\sum _2^P$-complete.