Extending regular edge-colorings of complete hypergraphs期刊界 All Journals 搜尽天下杂志传播学术成果专业期刊搜索期刊信息化学术搜索

Extending regular edge-colorings of complete hypergraphs

Authors:	Amin Bahmanian Sadegheh Haghshenas

Affiliation:	Department of Mathematics, Illinois State University, Normal, IIlinois

Abstract:	A coloring (partition) of the collection $(\binom{X}{h})$ of all $h$ -subsets of a set $X$ is $r$ -regular if the number of times each element of $X$ appears in each color class (all sets of the same color) is the same number $r$ . We are interested in finding the conditions under which a given $r$ -regular coloring of $(\binom{X}{h})$ is extendible to an $s$ -regular coloring of $(\binom{Y}{h})$ for $s ⩾ r$ and $Y ⊋ X$ . The case $◂,▸ h = 2, r = s = 1$ was solved by Cruse, and due to its connection to completing partial symmetric latin squares, many related problems are extensively studied in the literature, but very little is known for $h ⩾ 3$ . The case $r = s = 1$ was solved by Häggkvist and Hellgren, settling a conjecture of Brouwer and Baranyai. The cases $h = 2$ and $h = 3$ were solved by Rodger and Wantland, and Bahmanian and Newman, respectively. In this paper, we completely settle the cases $h = 4, ◂⩾▸ \| Y \| ⩾ 4 \| X \|$ and $h = 5, ◂⩾▸ \| Y \| ⩾ 5 \| X \|$ .

Keywords:	embedding factorization edge-coloring decomposition Baranyai's theorem amalgamation detachment