Extending edge‐colorings of complete hypergraphs into regular colorings

Abstract

Let true( 0X h true) be the collection of all h‐subsets of an n‐set X ⊇ Y. Given a coloring (partition) of a set S ⊆ true( 0.0ptX h true), we are interested in finding conditions under which this coloring is extendible to a coloring of true( 0X h true) so that the number of times each element of X appears in each color class (all sets of the same color) is the same number r. The case S = ∅ , r = 1 was studied by Sylvester in the 18th century and remained open until the 1970s. The case h = 2 , r = 1 is extensively studied in the literature and is closely related to completing partial symmetric Latin squares. For S = true( 0.0ptY h true), we settle the cases h = 4 , ∣ X ∣ ≥ 4.847323 ∣ Y ∣, and h = 5 , ∣ X ∣ ≥ 6.285214 ∣ Y ∣ completely. Moreover, we make partial progress toward solving the case where S = true( 0.0ptX h true) \ true( 0.0ptY h true). These results can be seen as extensions of the famous Baranyai’s theorem, and make progress toward settling a 40‐year‐old problem posed by Cameron.

Connected Fair Detachments of Hypergraphs

“…Embedding connected factorizations have been only studied for graphs [24,27,32,35]. For recent progress on embedding factorizations of hypergraphs, we refer the reader to [6,7,8,12,14,20]. In this section, we prove connected analogues of various hypergraph embedding results.…”

“…In this section, we prove connected analogues of various hypergraph embedding results. For the sake of brevity, we shall skip some of the algebraic manipulations that are similar to those in [6,7].…”

Extending regular edge‐colorings of complete hypergraphs

“…Coloring the edges of the forms u v { , } 3 and u v { , } 2 2 is rather involved, but coloring the remaining edges is relatively straightforward. At every step, we need to make sure that all the edges can actually be colored without potentially violating condition (2). We first color the edges of the form…”

“…The following shows that the coloring of the edges of the form u v { , } 3 satisfying (7) is possible. where the second to last equality is implied by (1 Now that we have colored all the edges of 4 , we shall show that our coloring satisfies (2). For For To prove the sufficiency of (B1) to (B3), let…”

“…For recent related results, we refer the reader to . Judging from the literature on embedding results for other combinatorial structures, finding a sharp lower bound for seems to be quite difficult.…”