On the Anti-Ramsey Number Under Edge Deletion
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arXiv
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2025
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| _version_ | 1866914193493458944 |
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| author | Ghalavand, Ali Jie, Qing Jin, Zemin Li, Xueliang Pan, Linshu |
| author_facet | Ghalavand, Ali Jie, Qing Jin, Zemin Li, Xueliang Pan, Linshu |
| contents | According to a study by Erdős et al. in 1975, the anti-Ramsey number of a graph \(G\), denoted as \(AR(n, G)\), is defined as the maximum number of colors that can be used in an edge-coloring of the complete graph \(K_n\) without creating a rainbow copy of \(G\). In this paper, we investigate the anti-Ramsey number under edge deletion and demonstrate that both decreasing and unchanging are possible outcomes. For three non-negative integers \(k\), \(t\), and \(n\), let \(G = kP_4 \cup tP_2\). Let \(E'\) be a subset of the edge set \(E(G)\) such that every endpoint of these edges has a degree of two in \(G\). We prove that if one of the conditions (i) \(t \geq k + 1 \geq 2\) and \(n \geq 8k + 2t - 4\); (ii) \(k, t \geq 1\) and \(n = 4k + 2t\); (iii) \(k = 1\), \(t \geq 1\), and \(n \geq 2t + 4\), occurs then the behavior of the anti-Ramsey number remains consistent when the edges in \(E'\) are removed from \(G\), i.e., \(AR(n, G) = AR(n, G - E')\). However, this is not the case when \(k \geq 2\), \(t = 0\), and \(n=4k\). As a result, we calculate \(AR(kP_4 \cup tP_2)\) for the cases: (i) \(t \geq k + 1 \geq 2\) and \(n \geq 8k + 2t - 4\); (ii) \(k, t \geq 1\) and \(n = 4k + 2t\); (iii) \(k = 1\), \(t \geq 0\), and \(n \geq 2t + 4\); (iv) \(k \geq 1\), \(t = 0\), and \(n = 4k\). |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2511_06034 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On the Anti-Ramsey Number Under Edge Deletion Ghalavand, Ali Jie, Qing Jin, Zemin Li, Xueliang Pan, Linshu Combinatorics According to a study by Erdős et al. in 1975, the anti-Ramsey number of a graph \(G\), denoted as \(AR(n, G)\), is defined as the maximum number of colors that can be used in an edge-coloring of the complete graph \(K_n\) without creating a rainbow copy of \(G\). In this paper, we investigate the anti-Ramsey number under edge deletion and demonstrate that both decreasing and unchanging are possible outcomes. For three non-negative integers \(k\), \(t\), and \(n\), let \(G = kP_4 \cup tP_2\). Let \(E'\) be a subset of the edge set \(E(G)\) such that every endpoint of these edges has a degree of two in \(G\). We prove that if one of the conditions (i) \(t \geq k + 1 \geq 2\) and \(n \geq 8k + 2t - 4\); (ii) \(k, t \geq 1\) and \(n = 4k + 2t\); (iii) \(k = 1\), \(t \geq 1\), and \(n \geq 2t + 4\), occurs then the behavior of the anti-Ramsey number remains consistent when the edges in \(E'\) are removed from \(G\), i.e., \(AR(n, G) = AR(n, G - E')\). However, this is not the case when \(k \geq 2\), \(t = 0\), and \(n=4k\). As a result, we calculate \(AR(kP_4 \cup tP_2)\) for the cases: (i) \(t \geq k + 1 \geq 2\) and \(n \geq 8k + 2t - 4\); (ii) \(k, t \geq 1\) and \(n = 4k + 2t\); (iii) \(k = 1\), \(t \geq 0\), and \(n \geq 2t + 4\); (iv) \(k \geq 1\), \(t = 0\), and \(n = 4k\). |
| title | On the Anti-Ramsey Number Under Edge Deletion |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2511.06034 |