Graceful coloring is computationally hard
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arXiv
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| Format: | Preprint |
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2024
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| author | Antony, Cyriac D., Laavanya S, Devi Yamini |
| author_facet | Antony, Cyriac D., Laavanya S, Devi Yamini |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2407_02179 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Graceful coloring is computationally hard Antony, Cyriac D., Laavanya S, Devi Yamini Combinatorics Discrete Mathematics 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. |
| title | Graceful coloring is computationally hard |
| topic | Combinatorics Discrete Mathematics |
| url | https://arxiv.org/abs/2407.02179 |