Graceful coloring is computationally hard

TL;DR

This work analyzes graceful coloring, a restricted labeling where a proper vertex coloring induces a proper edge coloring via difference labels. It establishes the fundamental bounds $χ(G^2) ≤ χ_g(G) ≤ a(χ(G^2))$ for every graph $G$ and identifies $χ_g(K_n)=a(n)$, tying graceful coloring to the OEIS sequence $a(n)$. The authors show substantial NP-hardness results for graceful coloring across several graph classes, including planar bipartite, regular, and 2-degenerate graphs, with specific results for $k ≥ 5$ and $k ≥ 6$, and prove NP-completeness for graceful 4-colorability in 2-degenerate graphs via a reduction from Positive Not-All-Equal $3$-Sat E4; the open case remains for planar graphs regarding graceful 4-colorability. Overall, the paper clarifies the computational difficulty of graceful coloring and situates it between distance-two coloring and integer-sequence objectives, highlighting both theoretical bounds and practical hardness implications.

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 $χ_g(G)$. We prove that $χ(G^2)\leq χ_g(G)\leq a(χ(G^2))$ 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.

Figures (5)

• Figure 1: A graceful coloring of a graph. This is a graceful 5-coloring.
• Figure 2: Variable gadget $G^X$. Vertex $x_j$ together with incident edges are replaced by this gadget, and the edges $e_1,e_2,e_3$ and $e_4$ incident on $x_j$ in $G_\mathcal{F}$ become the dashed edges of the gadget in $G$.
• Figure 3: Clause gadget $G^{C}$. Vertex $c_i$ together with incident edges are replaced by this gadget, and the edges $e_1,e_2$ and $e_3$ incident on $c_i$ in $G_\mathcal{F}$ become the dashed edges of the gadget in $G$.
• Figure 4: A graceful 4-coloring $h$ of $G^{X}$.
• Figure 5: Two graceful 4-colorings of $G^C$

Theorems & Definitions (8)

• Theorem 1: laavanya_deviYamini
• Theorem 2
• Theorem 3
• proof : Proof of Guarantee (overview)
• Theorem 4
• Theorem 5
• Theorem 6
• proof : Proof overview