Graceful coloring is computationally hard

TL;DR

This paper investigates the computational complexity of graceful colorings in graphs, establishing bounds on the graceful chromatic number and proving NP-hardness for various classes of graphs, highlighting the problem's difficulty.

Contribution

It provides bounds relating the chromatic number of a graph's square to its graceful chromatic number and proves NP-hardness for graceful coloring in multiple graph classes.

Findings

Bounds on $ ext{chi}_g(G)$ in terms of $ ext{chi}(G^2)$

NP-hardness of graceful coloring for planar bipartite, regular, and 2-degenerate graphs

NP-completeness for planar bipartite graphs with degree $k-2$ for $k extgreater{}5$

Abstract

Given a (proper) vertex coloring $f$ of a graph $G$, say $f\colon V(G)\to \mathbb{N}$, the difference edge labelling induced by $f$ is a function $h\colon E(G)\to \mathbb{N}$ defined as $h(uv)=|f(u)-f(v)|$ for every edge $uv$ of $G$. A graceful coloring of $G$ is a vertex coloring $f$ of $G$ such that the difference edge labelling $h$ induced by $f$ is a (proper) edge coloring of $G$. A graceful coloring with range $\{1,2,\dots,k\}$ is called a graceful $k$-coloring. The least integer $k$ such that $G$ admits a graceful $k$-coloring is called the graceful chromatic number of $G$, denoted by $\chi_g(G)$. We prove that $\chi(G^2)\leq \chi_g(G)\leq a(\chi(G^2))$ for every graph $G$, where $a(n)$ denotes the $n$th term of the integer sequence A065825 in OEIS. We also prove that graceful coloring problem is NP-hard for planar bipartite graphs, regular graphs and 2-degenerate graphs. In particular, we show that for each $k\geq 5$, it is NP-complete to check whether a planar bipartite graph of maximum degree $k-2$ is graceful $k$-colorable. The complexity of checking whether a planar graph is graceful 4-colorable remains open.