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| Main Authors: | , |
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| Format: | Preprint |
| Published: |
2024
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2408.10651 |
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| _version_ | 1866910571067080704 |
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| author | Lo, Allan Williams, Ella |
| author_facet | Lo, Allan Williams, Ella |
| contents | A classical result of Corrádi and Hajnal states that every graph $G$ on $n$ vertices with $n\in 3\mathbb{N}$ and $δ(G) \ge 2n/3$ contains a perfect triangle-tiling, i.e.,\ a spanning set of vertex-disjoint triangles. We explore a generalisation of this result to edge-coloured graphs. Let $G$ be an edge-coloured graph on $n$ vertices. The minimum colour degree $δ^c(G)$ of $G$ is the largest integer $k$ such that, for every vertex $v \in V(G)$, there are at least $k$ distinct colours on edges incident to $v$. We show that if $δ^c(G) \ge (5/6 + \varepsilon) n$, then $G$ has a spanning set of vertex-disjoint rainbow triangles. On the other hand, we find an example showing the bound should be at least $5n/7$. We also discuss a related tiling problems on digraphs, which may be of independent interest. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2408_10651 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Towards an edge-coloured Corrádi--Hajnal theorem Lo, Allan Williams, Ella Combinatorics A classical result of Corrádi and Hajnal states that every graph $G$ on $n$ vertices with $n\in 3\mathbb{N}$ and $δ(G) \ge 2n/3$ contains a perfect triangle-tiling, i.e.,\ a spanning set of vertex-disjoint triangles. We explore a generalisation of this result to edge-coloured graphs. Let $G$ be an edge-coloured graph on $n$ vertices. The minimum colour degree $δ^c(G)$ of $G$ is the largest integer $k$ such that, for every vertex $v \in V(G)$, there are at least $k$ distinct colours on edges incident to $v$. We show that if $δ^c(G) \ge (5/6 + \varepsilon) n$, then $G$ has a spanning set of vertex-disjoint rainbow triangles. On the other hand, we find an example showing the bound should be at least $5n/7$. We also discuss a related tiling problems on digraphs, which may be of independent interest. |
| title | Towards an edge-coloured Corrádi--Hajnal theorem |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2408.10651 |