A parameterized algorithm for $K_r$-factors in graphs of high minimum degree
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arXiv
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| Format: | Preprint |
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2023
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| _version_ | 1866917297359159296 |
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| author | Gan, Luyining Han, Jie Hu, Jie |
| author_facet | Gan, Luyining Han, Jie Hu, Jie |
| contents | A $K_r$-factor of a graph $G$ is a collection of vertex-disjoint $r$-cliques covering $V(G)$. We prove the following algorithmic version of the classical Hajnal--Szemerédi Theorem in graph theory, when $r$ is considered as a constant. Given $r, c, n\in \mathbb{N}$ such that $n\in r\mathbb N$, let $G$ be an $n$-vertex graph with minimum degree at least $(1-1/r)n - c$. Then there is an algorithm with running time $2^{c^{O(1)}} n^{O(1)}$ that outputs either a $K_r$-factor of $G$ or a certificate showing that none exists, namely, this problem is fixed-parameter tractable in $c$. On the other hand, it is known that if $c = n^{\varepsilon}$ for fixed $\varepsilon \in (0,1)$, the problem is \texttt{NP-C}. By taking the complement, our result yields a similar result on the equitable $Δ$-colorings of graphs of maximum degree $Δ+c$, for $Δ\in [n/r, n/(r-1)]$.
We indeed establish characterization theorems for this problem, showing that the existence of a $K_r$-factor is equivalent to the existence of certain class of $K_r$-tilings of size $o(n)$, whose existence can be searched by the color-coding technique developed by Alon--Yuster--Zwick. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2307_08056 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | A parameterized algorithm for $K_r$-factors in graphs of high minimum degree Gan, Luyining Han, Jie Hu, Jie Combinatorics Computational Complexity A $K_r$-factor of a graph $G$ is a collection of vertex-disjoint $r$-cliques covering $V(G)$. We prove the following algorithmic version of the classical Hajnal--Szemerédi Theorem in graph theory, when $r$ is considered as a constant. Given $r, c, n\in \mathbb{N}$ such that $n\in r\mathbb N$, let $G$ be an $n$-vertex graph with minimum degree at least $(1-1/r)n - c$. Then there is an algorithm with running time $2^{c^{O(1)}} n^{O(1)}$ that outputs either a $K_r$-factor of $G$ or a certificate showing that none exists, namely, this problem is fixed-parameter tractable in $c$. On the other hand, it is known that if $c = n^{\varepsilon}$ for fixed $\varepsilon \in (0,1)$, the problem is \texttt{NP-C}. By taking the complement, our result yields a similar result on the equitable $Δ$-colorings of graphs of maximum degree $Δ+c$, for $Δ\in [n/r, n/(r-1)]$. We indeed establish characterization theorems for this problem, showing that the existence of a $K_r$-factor is equivalent to the existence of certain class of $K_r$-tilings of size $o(n)$, whose existence can be searched by the color-coding technique developed by Alon--Yuster--Zwick. |
| title | A parameterized algorithm for $K_r$-factors in graphs of high minimum degree |
| topic | Combinatorics Computational Complexity |
| url | https://arxiv.org/abs/2307.08056 |