Chromatic and achromatic numbers of unitary addition Cayley graphs
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| Format: | Preprint |
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2024
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| _version_ | 1866909595078754304 |
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| author | Calhoun, Keenan Karabulut, Yeşim Demiroğlu Pigno, Vincent Timmons, Craig |
| author_facet | Calhoun, Keenan Karabulut, Yeşim Demiroğlu Pigno, Vincent Timmons, Craig |
| contents | Let $R$ be a ring. The unitary addition Cayley graph of $R$, denoted $\mathcal{U}(R)$, is the graph with vertex $R$, and two distinct vertices $x$ and $y$ are adjacent if and only if $x+y$ is a unit. We determine a formula for the clique number and chromatic number of such graphs when $R$ is a finite commutative ring with an odd number of elements. This includes the special case when $R$ is $\mathbb{Z}_n$, the integers modulo $n$, where these parameters had been found under the assumption that $n$ is even, or $n$ is a power of an odd prime. Additionally, we study the achromatic number of $\mathcal{U}( \mathbb{Z}_n )$ in the case that $n$ is the product of two primes. We prove that the achromatic number of $\mathcal{U} ( \mathbb{Z}_{3q})$ is equal to $\frac{3q+1}{2}$ when $q > 3$ is a prime. We also prove a lower bound that applies when $n = pq$ where $p$ and $q$ are distinct odd primes. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2407_10364 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Chromatic and achromatic numbers of unitary addition Cayley graphs Calhoun, Keenan Karabulut, Yeşim Demiroğlu Pigno, Vincent Timmons, Craig Combinatorics Let $R$ be a ring. The unitary addition Cayley graph of $R$, denoted $\mathcal{U}(R)$, is the graph with vertex $R$, and two distinct vertices $x$ and $y$ are adjacent if and only if $x+y$ is a unit. We determine a formula for the clique number and chromatic number of such graphs when $R$ is a finite commutative ring with an odd number of elements. This includes the special case when $R$ is $\mathbb{Z}_n$, the integers modulo $n$, where these parameters had been found under the assumption that $n$ is even, or $n$ is a power of an odd prime. Additionally, we study the achromatic number of $\mathcal{U}( \mathbb{Z}_n )$ in the case that $n$ is the product of two primes. We prove that the achromatic number of $\mathcal{U} ( \mathbb{Z}_{3q})$ is equal to $\frac{3q+1}{2}$ when $q > 3$ is a prime. We also prove a lower bound that applies when $n = pq$ where $p$ and $q$ are distinct odd primes. |
| title | Chromatic and achromatic numbers of unitary addition Cayley graphs |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2407.10364 |