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Distribution of colors in Gallai colorings
- Source :
- European Journal of Combinatorics, 86 (2020) 103087
- Publication Year :
- 2019
-
Abstract
- A Gallai coloring is an edge coloring that avoids triangles colored with three different colors. Given integers $e_1\ge e_2 \ge \dots \ge e_k$ with $\sum_{i=1}^ke_i={n \choose 2}$ for some $n$, does there exist a Gallai $k$-coloring of $K_n$ with $e_i$ edges in color $i$? In this paper, we give several sufficient conditions and one necessary condition to guarantee a positive answer to the above question. In particular, we prove the existence of a Gallai-coloring if $e_1-e_k\le 1$ and $k \le \lfloor n/2\rfloor$. We prove that for any integer $k\ge 3$ there is a (unique) integer $g(k)$ with the following property: there exists a Gallai $k$-coloring of $K_n$ with $e_i$ edges in color $i$ for every $e_1\le\dots \le e_k$ satisfying $\sum_{i=1}^ke_i={n\choose 2}$, if and only if $n\ge g(k)$. We show that $g(3)=5$, $g(4)=8$, and $2k-2\le g(k)\le 8k^2+1$ for every $k\ge 3$.
- Subjects :
- Mathematics - Combinatorics
Subjects
Details
- Database :
- arXiv
- Journal :
- European Journal of Combinatorics, 86 (2020) 103087
- Publication Type :
- Report
- Accession number :
- edsarx.1903.04380
- Document Type :
- Working Paper
- Full Text :
- https://doi.org/10.1016/j.ejc.2020.103087