Cycles of weight divisible by $k$
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866916308065452032 |
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| author | Diwan, Ajit A. |
| author_facet | Diwan, Ajit A. |
| contents | A weighted (directed) graph is a (directed) graph with integer weights assigned to its vertices and edges. The weight of a subgraph is the sum of weights of vertices and edges in the subgraph. The problem of determining the largest order $f(k)$ of a weighted complete directed graph that does not contain a directed cycle of weight divisible by $k$, for an integer $k \ge 2$, was raised by Alon and Krivelevich [J. Graph Theory 98 (2021) 623-629]. They showed that $f(k)$ is $O(k\log k)$ and $f(k) \le 2k-2$ if $k$ is prime. The best bounds known to us are $f(k) \le 2k-2$ for all $k$ and $f(k) < (3k-1)/2$ for prime $k$. It is also known that $f(k) \ge k$ and this is believed to be the correct value. We prove that $f(k) < k+2Ω(k)$, where $Ω(k)$ is the number of prime factors, not necessarily distinct, in the prime factorization of $k$.
We also show that any weighted undirected graph of minimum degree $2k-1$ contains a cycle of weight divisible by $k$. This result is proved in the more general setting in which the weights are from a finite abelian group of order $k$, and the cycle has weight equal to the group identity. We conjecture that this holds for undirected graphs with minimum degree $k+1$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2407_01198 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Cycles of weight divisible by $k$ Diwan, Ajit A. Combinatorics Discrete Mathematics 05C35 05C22 05C38 A weighted (directed) graph is a (directed) graph with integer weights assigned to its vertices and edges. The weight of a subgraph is the sum of weights of vertices and edges in the subgraph. The problem of determining the largest order $f(k)$ of a weighted complete directed graph that does not contain a directed cycle of weight divisible by $k$, for an integer $k \ge 2$, was raised by Alon and Krivelevich [J. Graph Theory 98 (2021) 623-629]. They showed that $f(k)$ is $O(k\log k)$ and $f(k) \le 2k-2$ if $k$ is prime. The best bounds known to us are $f(k) \le 2k-2$ for all $k$ and $f(k) < (3k-1)/2$ for prime $k$. It is also known that $f(k) \ge k$ and this is believed to be the correct value. We prove that $f(k) < k+2Ω(k)$, where $Ω(k)$ is the number of prime factors, not necessarily distinct, in the prime factorization of $k$. We also show that any weighted undirected graph of minimum degree $2k-1$ contains a cycle of weight divisible by $k$. This result is proved in the more general setting in which the weights are from a finite abelian group of order $k$, and the cycle has weight equal to the group identity. We conjecture that this holds for undirected graphs with minimum degree $k+1$. |
| title | Cycles of weight divisible by $k$ |
| topic | Combinatorics Discrete Mathematics 05C35 05C22 05C38 |
| url | https://arxiv.org/abs/2407.01198 |