Graceful coloring is computationally hard

• Published in: arXiv.org
• Main Authors: Cyriac Antony, Laavanya, D, Devi, Yamini S
• Format: Paper
• Language: English
• Published: Ithaca Cornell University Library, arXiv.org 22.07.2024
• Subjects: Graph coloring; Graph theory; Graphs; Integers; Labeling
• ISSN: 2331-8422

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.