On the algorithmic complexity of adjacent vertex closed distinguishing colorings number of graphs

TL;DR

This paper investigates the complexity and bounds of the closed distinguishing number and list coloring of graphs, introduces new bounds, and proves NP-completeness for certain decision problems related to these colorings.

Contribution

It establishes that the difference between the closed distinguishing number and list version can be arbitrarily large, improves upper bounds using Nullstellensatz, and proves NP-completeness for specific graph classes.

Findings

Existence of bipartite graphs with arbitrarily large dis[G]

Dis[G] can be significantly less than dis_ell[G]

NP-completeness results for planar subcubic graphs and general graphs

Abstract

An assignment of numbers to the vertices of graph G is closed distinguishing if for any two adjacent vertices v and u the sum of labels of the vertices in the closed neighborhood of the vertex v differs from the sum of labels of the vertices in the closed neighborhood of the vertex u unless they have the same closed neighborhood (i.e. N[u]=N[v]). The closed distinguishing number of G, denoted by dis[G], is the smallest integer k such that there is a closed distinguishing labeling for G using integers from the set[k].Also, for each vertex $v \in V(G)$, let L(v) denote a list of natural numbers available at v. A list closed distinguishing labeling is a closed distinguishing labeling f such that $f(v)\in L(v)$ for each $v \in V(G)$.A graph G is said to be closed distinguishing k-choosable if every k-list assignment of natural numbers to the vertices of G permits a list closed distinguishing labeling of G. The closed distinguishing choice number of G, $dis_{\ell}[G]$, is the minimum number k such that G is closed distinguishing k-choosable. We show that for each integer t there is a bipartite graph G such that $dis[G] > t$.It was shown that for every graph G with $\Delta\geq 2$, $dis[G]\leq dis_{\ell}[G]\leq \Delta^2-\Delta+1$ and there are infinitely values of $\Delta$ for which G might be chosen so that $dis[G] =\Delta^2-\Delta+1$. We show that the difference between $dis[G]$ and $dis_{\ell}[G]$ can be arbitrary large and for every positive integer t there is a graph G such that $dis_{\ell}[G]-dis[G]\geq t$. We improve the current upper bound and give some number of upper bounds for the closed distinguishing choice number by using the Combinatorial Nullstellensatz. We show that it is $\mathbf{NP}$-complete to decide for a given planar subcubic graph G, whether dis[G]=2. Also, we prove that for every $k\geq 3$, it is {\bf NP}-complete to decide whether $dis[G]=k$ for a given graph G