Extending Edge-colorings of Complete Hypergraphs into Regular Colorings
EEXTENDING EDGE-COLORINGS OF COMPLETE HYPERGRAPHSINTO REGULAR COLORINGS
AMIN BAHMANIAN
Abstract.
Let ` Xh ˘ be the collection of all h -subsets of an n -set X Ě Y . Given acoloring (partition) of a set S Ď ` Xh ˘ , we are interested in finding conditions underwhich this coloring is extendible to a coloring of ` Xh ˘ so that the number of timeseach element of X appears in each color class (all sets of the same color) is the samenumber r . The case S “ ∅ , r “ h “ , r “ S “ ` Yh ˘ , we settle the cases h “ , | X | ě . | Y | , and h “ , | X | ě . | Y | completely. Moreover, we make partial progress toward solving the casewhere S “ ` Xh ˘ z ` Yh ˘ . These results can be seen as extensions of the famous Baranyai’stheorem, and make progress toward settling a 40-year-old problem posed by Cameron. Introduction
Suppose that we have been entrusted to color (or partition) the collection ` r n s h ˘ ofall h -subsets of the n -set r n s : “ t , . . . , n u so that the number of times each element of r n s appears in each color class (all sets of the same color) is exactly r . Such a coloringis called an r -factorization of ` r n s h ˘ . A solution for the case n “ , h “ , r “ t , , u , t , , u t , , u , t , , u t , , u , t , , u t , , u , t , , u t , , u , t , , ut , , u , t , , u t , , u , t , , u t , , u , t , , u t , , u , t , , u t , , u , t , , u Note that the number of times each element of r n s appears in ` r n s h ˘ is ` n ´ h ´ ˘ . Thus,for ` r n s h ˘ to be r -factorable, it is clear that (i) r must divide ` n ´ h ´ ˘ . In addition, asimple double counting argument shows that (ii) h must divide rn . One may wonderif conditions (i) and (ii) are also sufficient for ` r n s h ˘ to be r -factorable. In the 18thcentury, Sylvester considered the case r “ ` r n s h ˘ is r -factorable if and only if h | rn and r | ` n ´ h ´ ˘ .We are interested in a Sudoku-type version of Baranyai’s theorem. A partial r -factorization of a set S Ď ` r n s h ˘ is a coloring of S with at most ` n ´ h ´ ˘ { r colors so that Mathematics Subject Classification.
Key words and phrases. embedding, factorization, edge-coloring, decomposition, Baranyai’s theo-rem, amalgamation, detachment. a r X i v : . [ m a t h . C O ] F e b XTENDING EDGE-COLORINGS OF COMPLETE HYPERGRAPHS 2 the number of times each element of r n s appears in each color class is at most r . Notethat a color class may be empty. Problem 1.
Under what conditions can a partial r -factorization of S Ď ` r n s h ˘ be ex-tended to an r -factorization of ` r n s h ˘ ? We are given a coloring of a subset S Ď ` r n s h ˘ , and our task is to complete the col-oring. In other words, we need to color T : “ ` r n s h ˘ z S so that the coloring of S Y T provides an r -factorization of ` r n s h ˘ . Baranyai’s theorem settles the case when S “ ∅ .A partial 4-factorization of ` r s ˘ is given below (Here we abbreviate a set t a, b, c u to abc ).156, 248, 379, 126, 348, 579, 127, 349, 568, 124, 389, 567148, 267, 359, 168, 279, 345, 159, 278, 346, 134, 259128, 347, 569, 178, 249, 356, 169, 247, 358, 123146, 239, 578, 137, 289, 456, 136, 257129, 367, 458, 125, 368, 479, 147, 258, 369, 157189, 246, 357, 158, 237, 469, 138, 245, 679, 139, 268145, 236, 789, 167, 238, 459, 149, 256, 378, 135, 269, 478It is not too difficult to extend this to the following 4-factorization.156, 248, 379, 126, 348, 579, 127, 349, 568, 124, 389, 567148, 267, 359, 168, 279, 345, 159, 278, 346, 134, 259, 678128, 347, 569, 178, 249, 356, 169, 247, 358, 123, 467, 589146, 239, 578, 137, 289, 456, 136, 257, 489, 179, 235, 468129, 367, 458, 125, 368, 479, 147, 258, 369, 157, 234, 689189, 246, 357, 158, 237, 469, 138, 245, 679, 139, 268, 457145, 236, 789, 167, 238, 459, 149, 256, 378, 135, 269, 478The case h “ , r “ r “ ` r m s h ˘ for some m ă n , was studiedby Cruse (for h “
2) [8], Cameron [7], and Baranyai and Brouwer [6]. Baranyai andBrouwer conjectured that a 1-factorization of ` r m s h ˘ can be extended to a 1-factorizationof ` r n s h ˘ if and only if n ě m and h divides m, n . H¨aggkvist and Hellgren [10] gave abeautiful proof of this conjecture. For further generalizations of H¨aggkvist-Hellgren’sresult, we refer the reader to two recent papers by the author and Newman [2, 3]in which extending r -factorizations of ` r m s h ˘ to s -factorizations of ` r n s h ˘ is studied (for s ě r ).At this point, it should be clear to the reader that the 1-factorization of ` r s ˘ in thefirst example, can not be extended to a 1-factorization of ` r s ˘ , but it can be extendedto a 1-factorization of ` r s ˘ .Like most results in the literature, our primary focus is the case where S “ ` r m s h ˘ (for some m ă n ). However, unlike those, here we do not require the given partial XTENDING EDGE-COLORINGS OF COMPLETE HYPERGRAPHS 3 factorization to be a factorization itself. In this case, Problem 1 was settled by Rodgerand Wantland over 20 years ago for h “ h “ , n ě . m [4]. In this paper, we settle the cases h “ , n ě . m and h “ , n ě . m . The major obstacle from h “ h ě r -factorization of ` r m s h ˘ to an r -factorizationof ` r n s h ˘ (for n ě m ), it is clearly necessary that r | ` n ´ h ´ ˘ , h | rn . Let χ p m, h, r q be thesmallest n such that any partial r -factorization of ` r m s h ˘ satisfying r | ` n ´ h ´ ˘ , h | rn canbe extended to an r -factorization of ` r n s h ˘ . Combining the results of this paper withthose of [2, 3, 4], it can be easily shown that 2 m ď χ p m, , r q ď . m, m ď χ p m, , r q ď . m , and 2 m ď χ p m, , r q ď . m .Last but not least, we shall consider Problem 1 in the case when S “ ` r n s h ˘ z ` r m s h ˘ . Inthis direction, we solve a variation of the problem when we allow sets of size less than h , and in our extension of the coloring we also extend the sets of size less than h tosets of size h .The paper is self-contained and all the preliminaries are given in Section 2. In section3, we shall consider Problem 1 in the case when S “ ` r n s h ˘ z ` r m s h ˘ . The cases h “ , Notation and Tools A hypergraph G is a pair p V p G q , E p G qq where V p G q is a finite set called the vertex set, E p G q is the edge multiset, where every edge is itself a multi-subset of V p G q . This meansthat not only can an edge occur multiple times in E p G q , but also each vertex can havemultiple occurrences within an edge. By an edge of the form t u m , u m , . . . , u m s s u ,we mean an edge in which vertex u i occurs m i times for 1 ď i ď r . The totalnumber of occurrences of a vertex v among all edges of E p G q is called the degree ,deg G p v q of v in G . The multiplicity of an edge e in G , written mult G p e q , is thenumber of repetitions of e in E p G q (note that E p G q is a multiset, so an edge mayappear multiple times). If t u m , u m , . . . , u m s s u is an edge in G , then we abbreviatemult G pt u m , u m , . . . , u m s s uq to mult G p u m , u m , . . . , u m s s q . If U , . . . , U s are multi-subsetsof V p G q , then mult G p U , . . . , U s q means mult G p Ť si “ U i q , where the union of U i s is theusual union of multisets. Whenever it is not ambiguous, we drop the subscripts; forexample we write deg p v q and mult p e q instead of deg G p v q and mult G p e q , respectively.For h P N , G is said to be h - uniform if | e | “ h for each e P E , and an h -factor in ahypergraph G is a spanning h -regular sub-hypergraph. An h -factorization is a partitionof the edge set of G into h -factors. The hypergraph K hn : “ p V, ` Vh ˘ q with | V | “ n is calleda complete h -uniform hypergraph. A k -edge-coloring of G is a mapping f : V p G q Ñ r k s and color class i of G , written G p i q , is the sub-hypergraph of G induced by the edgesof color i .Let G be a hypergraph, let U be some finite set, and let Ψ : V p G q Ñ U be asurjective mapping. The map Ψ extends naturally to E p G q . For A P E p G q we defineΨ p A q “ t Ψ p x q : x P A u . Note that Ψ need not be injective, and A may be a multiset. XTENDING EDGE-COLORINGS OF COMPLETE HYPERGRAPHS 4
Then we define the hypergraph F by taking V p F q “ U and E p F q “ t Ψ p A q : A P E p G qu . We say that F is an amalgamation of G , and that G is a detachment of F .Associated with Ψ is a (number) function g defined by g p u q “ | Ψ ´ p u q| ; to be morespecific we will say that G is a g -detachment of F . Then G has ř u P V p F q g p u q vertices.Note that Ψ induces a bijection between the edges of F and the edges of G , and thatthis bijection preserves the size of an edge. We adopt the convention that it preservesthe color also, so that if we amalgamate or detach an edge-colored hypergraph theamalgamation or detachment preserves the same coloring on the edges. We makeexplicit a straightforward observation: Given G , V p F q and Ψ the amalgamation isuniquely determined, but given F , V p G q and Ψ the detachment is in general far fromuniquely determined.There are quite a lot of other papers on amalgamations and some highlights include[9, 11, 12, 13, 14, 15, 17, 18].Given an edge-colored hypergraph F , we are interested in finding a detachment G obtained by splitting each vertex of F into a prescribed number of vertices in G sothat (i) the degree of each vertex in each color class of F is shared evenly among thesubvertices in the same color class in G , and (ii) the multiplicity of each edge in F isshared evenly among the subvertices in G . The following theorem, which is a specialcase of a general result in [1], guarantees the existence of such detachment (Here x « y means t y u ď x ď r y s ). Theorem 2.1. (Bahmanian [1, Theorem 4.1])
Let F be a k -edge-colored hypergraphand let g : V p F q Ñ N . Then there exists a g -detachment G (possibly with multipleedges) of F whose edges are all sets, with amalgamation function Ψ : V p G q Ñ V p F q , g being the number function associated with Ψ , such that (F1) for each u P V p F q , each v P Ψ ´ p u q and i P r k s , deg G p i q p v q « deg F p i q p u q g p u q ;(F2) for distinct u , . . . , u s P V p F q and U i Ď Ψ ´ p u i q with | U i | “ m i ď g p u i q for i P r s s , mult G p U , . . . , U s q « mult F p u m , . . . , u m s s q Π si “ ` g p u i q m i ˘ . Let Ą K hm be the hypergraph obtained by adding a new vertex u and new edges to K hm so thatmult p u i , W q “ ˆ n ´ mi ˙ for each i P r h s , and W Ď V p K hm q with | W | “ h ´ i. In other words, Ą K hm is an amalgamation of K hn , obtained by identifying an arbitraryset of n ´ m vertices in K hn .An immediate consequence of Theorem 2.1 is the following. Corollary 2.2.
Let k : “ ` n ´ h ´ ˘ { r P N . A partial r -factorization of K hm can be extendedto an r -factorization of K hn if and only if the new edges of F : “ Ą K hm can be colored so XTENDING EDGE-COLORINGS OF COMPLETE HYPERGRAPHS 5 that (1) @ i P r k s deg F p i q p v q “ " r if v ‰ u,r p n ´ m q if v “ u. Proof.
First, suppose that a partial r -factorization of K hm can be extended to an r -factorization of K hn . By amalgamating the new n ´ m vertices of K hn into a singlevertex u , we clearly obtain F . The k -edge-coloring of K hn (in which each color class isan r -factor) induces a k -edge-coloring in F that satisfies (1).Conversely, suppose that the edges of F are colored so that (1) is satisfied. Let g : V p F q Ñ N with g p u q “ n ´ m , and g p v q “ v ‰ u . By Theorem 2.1, thereexists a g -detachment G of F such that(a) for each v P Ψ ´ p u q , and i P r k s deg G p i q p v q « deg F p i q p u q{ g p u q “ r p n ´ m q{p n ´ m q “ r. (b) for U Ď Ψ ´ p u q , W Ď V p K hm q with | U | “ i, | W | “ h ´ i , for i P r h s .mult G p U, W q « mult F p u i , W q ` g p u q i ˘ “ ` n ´ mi ˘` n ´ mi ˘ “ r -factor, and by (b), G – K hn . (cid:3) The following observation will be quite useful throughout the paper.
Proposition 2.3.
For every n, m, h P N with n ě m ě h , (2) ˆ nh ˙ “ h ÿ i “ ˆ mi ˙ˆ n ´ mh ´ i ˙ . (3) m r ˆ n ´ h ´ ˙ ´ ˆ m ´ h ´ ˙ s “ h ´ ÿ i “ i ˆ mi ˙ˆ n ´ mh ´ i ˙ . Proof.
The proof of (2) is straightforward. Let F be a hypergraph with vertex set t u, v u such that mult p u i , v h ´ i q “ ` mi ˘` n ´ mh ´ i ˘ for 0 ď i ď h ´
1. Note that F is anamalgamation of the hypergraph G with edge set ` Xh ˘ z ` Uh ˘ where | X | “ n, | U | “ m .Double counting the degree of u proves (3): h ´ ÿ i “ i ˆ mi ˙ˆ n ´ mh ´ i ˙ “ deg F p u q “ ÿ u P U d G p u q “ m r ˆ n ´ h ´ ˙ ´ ˆ m ´ h ´ ˙ s . (cid:3) In order to avoid trivial cases, throughout the rest of this paper we assume that m ą h . XTENDING EDGE-COLORINGS OF COMPLETE HYPERGRAPHS 6 Arbitrary h If we replace every edge e of a hypergraph G by λ copies of e , then we denote thenew hypergraph by λ G . For hypergraphs G , . . . , G t with the same vertex set V , wedefine their union , written Ť ti “ G i , to be the hypergraph with vertex set V and edgeset Ť ti “ E p G i q . For a hypergraph G and V Ď V p G q , let G ´ V be the hypergraph whosevertex set is V p G qz V and whose edge set is t e z V | e P E p G qu .Let V be an arbitrary subset of vertices in K hn with | V | “ m ď n . Then K hn ´ V – Ť h ´ i “ ` mi ˘ K h ´ in ´ m . A partial r -factorization of H : “ K hn ´ V is a coloring of the edges of K hn ´ V with at most ` n ´ h ´ ˘ { r colors so that for each color i , deg H p i q p v q ď r for eachvertex of H (Note that H has singleton edges). In the next result, we completely settlethe problem of extending a partial r -factorization of K hn ´ V to an r -factorization of K hn . Note that here we are not only extending the coloring, but also the edges of sizeless than h to edges of size h . The case h “ Theorem 3.1.
For V Ď V p K hn q with | V | “ m , any partial r -factorization of H : “ K hn ´ V can be extended to an r -factorization of K hn if and only if h | rn , r | ` n ´ h ´ ˘ , andfor all i “ , , . . . , ` n ´ h ´ ˘ { r , (4) d H p i q p v q “ r @ v P V p H q , (5) | E p H p i qq| ď rnh . Proof.
To prove the necessity, suppose that a given partial r -factorization of H isextended to an r -factorization of K hn . For K hn to be r -factorable, the two divisibilityconditions are clearly necessary. By extending an edge e of size i ( i ă h ) in H to anedge of size h in K hn , the color of e does not change, and so (4) is necessary. Sincethe number of edges in each color class of K hn is exactly rn { h , the necessity of (5) isimplied.To prove the sufficiency, suppose that a partial r -factorization of H is given, h | rn , r | ` n ´ h ´ ˘ , and that (4), (5) are satisfied. Let k “ ` n ´ h ´ ˘ , and let F “ Č K hn ´ m . For0 ď i ď h , an edge of type u i in F is an edge in F containing u i but not containing u i ` . Note that there are ` mi ˘` n ´ mh ´ i ˘ edges of type u i in F .There is a clear one-to-one correspondence between the edges of size i in H and theedges of type u h ´ i in F for each i P r h s . We color the edges of type u i in F with thesame color as the corresponding edge in H for 0 ď i ď h ´
1. By Corollary 2.2, if wecan color the remaining edges of F (edges of type u h ) so that the following conditionis satisfied, then we are done.(6) @ i P r k s deg F p i q p v q “ " r if v ‰ u,rm if v “ u. Let mult i p u j , . q be the number of edges of type u j in F p i q , for i P r k s , j P r h s . Notethat mult i p u h , . q “ mult F p i q p u h q for i P r k s . We color the edges of type u h so that for XTENDING EDGE-COLORINGS OF COMPLETE HYPERGRAPHS 7 i P r k s , mult i p u h , . q “ rnh ´ r p n ´ m q ` h ´ ÿ j “ j mult i p u h ´ j ´ , . q . Since h | rn , mult i p u h , . q is an integer for i P r k s . The following shows that mult i p u h , . q ě i P r k s . rnh p q ě | E p H p i qq| “ h ´ ÿ j “ mult i p u j , . q“ h ÿ j “ j mult i p u h ´ j , . q ´ h ´ ÿ j “ j mult i p u h ´ j ´ , . q p q “ r p n ´ m q ´ h ´ ÿ j “ j mult i p u h ´ j ´ , . q . Now we show that all edges of the type u h will be colored, or equivalently that, ř ki “ mult i p u h , . q “ ` mh ˘ . k ÿ i “ mult i p u h , . q “ k ÿ i “ ` rnh ´ r p n ´ m q ` h ´ ÿ j “ j mult i p u h ´ j ´ , . q ˘ “ rknh ´ rk p n ´ m q ` h ´ ÿ j “ j k ÿ i “ mult i p u h ´ j ´ , . q“ ˆ nh ˙ ´ p n ´ m q ˆ n ´ h ´ ˙ ` h ÿ j “ p j ´ q ˆ mh ´ j ˙ˆ n ´ mj ˙ p q , p q “ h ÿ j “ ˆ mj ˙ˆ n ´ mh ´ j ˙ ´ h ´ ÿ j “ j ˆ n ´ mj ˙ˆ mh ´ j ˙ ´ p n ´ m q ˆ n ´ m ´ h ´ ˙ ` h ÿ j “ p j ´ q ˆ mh ´ j ˙ˆ n ´ mj ˙ “ ˆ mh ˙ ´ p n ´ m q ˆ n ´ m ´ h ´ ˙ ` h ˆ n ´ mh ˙ “ ˆ mh ˙ . XTENDING EDGE-COLORINGS OF COMPLETE HYPERGRAPHS 8
To complete the proof, we show that deg F p i q p u q “ rm for i P r k s . We havedeg F p i q p u q “ h ÿ j “ j mult i p u j , . q “ h mult i p u h , . q ` h ÿ j “ p h ´ j q mult i p u h ´ j , . q“ h mult i p u h , . q ` h h ÿ j “ mult i p u h ´ j , . q ´ h ÿ j “ j mult i p u h ´ j , . q“ h h ÿ j “ mult i p u h ´ j , . q ´ h ÿ j “ j mult i p u h ´ j , . q“ rn ´ r p n ´ m q “ rm. (cid:3) For a hypergraph G and V Ď V p G q , let G z V be the hypergraph whose vertex set is V p G q and whose edge set is t e P E p G q| e Ę V u .Let V Ď V p K hn q with | V | “ m ď n , and let H : “ K hn z V . An edge e P E p H q is of type i , if | e X V | “ i (for 0 ď i ď h ´ P be a partial r -factorization of H . Thena partial r -factorization Q of H is said to be P -friendly if(a) the color of each edge of type 0 is the same in P and Q , and(b) the number of edges of type i and color j is the same in P and Q for each i P r h ´ s and each color j .We are interested in finding the conditions under which a partial r -factorization of H can be extended to an r -factorization of K hn . Lemma 3.2.
For V Ď V p K hn q with | V | “ m , if a partial r -factorization of H : “ K hn z V can be extended to an r -factorization of K hn , then(N1) h | rn ,(N2) r | ` n ´ h ´ ˘ ,(N3) d H p i q p v q “ r for each v P V p H qz V , and i P r k s ,(N4) | E p H p i qq| ď rn { h for i P r k s ,where k : “ ` n ´ h ´ ˘ { r . It remains an open question whether these conditions are sufficient. Here we provea weaker result.
Corollary 3.3.
Let V Ď V p K hn q with | V | “ m , and let P be a partial r -factorization of H : “ K hn z V , and assume that (N1)–(N4) are satisfied. Then there exists a P -friendlypartial r -factorization of H that can be extended to an r -factorization of K hn .Proof. By eliminating all the vertices in V , and shrinking the edges containing verticesin V , we obtain K hn ´ V . The rest of the proof follows from Theorem 3.1. (cid:3) h “ Theorem 4.1.
For n ě . m , any partial r -factorization of K m can be extendedto an r -factorization of K n if and only if | rn and r | ` n ´ ˘ . XTENDING EDGE-COLORINGS OF COMPLETE HYPERGRAPHS 9
Proof.
For the necessary conditions, see the previous section. To prove the sufficiency,we need to show that the edges of F : “ Ą K m can be colored with k : “ ` n ´ ˘ { r colors sothat (6) is satisfied.First we color the edges in F of the form W Y t u u where W Ď V : “ V p K m q and | W | “
3. We color these edges greedily so that deg i p x q ď r for each x P V and i P r k s .We claim that this coloring can be done in such a way that all edges of this type arecolored. Suppose by contrary that there is an edge in F of the form t x, y, z, u u with x, y, z P V that can not be colored. This implies that for each i P r k s either deg i p x q “ r or deg i p y q “ r or deg i p z q “ r . Thus for each i P r k s , deg i p x q ` deg i p y q ` deg i p z q ě r .On the one hand, ř ki “ ` deg i p x q ` deg i p y q ` deg i p z q ˘ ě rk “ ` n ´ ˘ , and on the otherhand, ř ki “ ` deg i p x q ` deg i p y q ` deg i p z q ˘ ď r ` m ´ ˘ ` p n ´ m q ` m ´ ˘ ´ s . Thus, wehave 3 r ˆ m ´ ˙ ` p n ´ m q ˆ m ´ ˙ ´ s ě ˆ n ´ ˙ . which is equivalent to f p n, m q : “ n ´ n ´ m n ` mn ´ n ` m ´ m ´ m ` ď
0. Now, we show that since n ą m and m ě
5, we have f p n, m q ą
0, which is acontradiction, and therefore, all edges in F of the form W Y t u u where W Ď V and | W | “ m ě
5, both 7 m ` m ´ m ´ m ´ f p n, m q “ n ´ n p n ´ q ´ m ` m ´ ¯ ` m p m ´ m ´ q ` ą n ´ m p m ´ q ´ m ` m ´ ¯ ` m p m ´ m ´ q ` “ n p m ` m ´ q ` m p m ´ m ´ q ` ą . Now we greedily color all the edges of the form W Y t u u where W Ď V and | W | “ i p x q ď r for each x P V and i P r k s . We show that this is possible. If bycontrary, some edge t x, y, u u with x, y P V remains uncolored, then for each i P r k s either deg i p x q “ r or deg i p y q “ r , and so deg i p x q ` deg i p y q ě r . We have ˆ n ´ ˙ “ rk ď k ÿ i “ ` deg i p x q ` deg i p y q ˘ ď r ˆ m ´ ˙ ` p n ´ m q ˆ m ´ ˙ ` p m ´ q ˆ n ´ m ˙ ´ s , which is equivalent to n ´ mn ` m n ` mn ´ n ´ m ´ m ´ m ` ď
0. UsingMathematica (Wolfram Alpha) it can be shown that this inequality does not have anyreal solution under the constraints m ě , n ě . m . Therefore, all edges of theform W Y t u u where W Ď V and | W | “
2, can be colored.
XTENDING EDGE-COLORINGS OF COMPLETE HYPERGRAPHS 10
Since for each x P V , k ÿ i “ ` r ´ deg i p x q ˘ “ rk ´ r ˆ m ´ ˙ ` p n ´ m q ˆ m ´ ˙ ` p m ´ q ˆ n ´ m ˙ s“ ˆ n ´ ˙ ´ ˆ m ´ ˙ ´ p n ´ m q ˆ m ´ ˙ ´ p m ´ q ˆ n ´ m ˙ p q “ ˆ n ´ m ˙ , we can color all the edges of the form t w, u u where w P V so that for each x P V ,there are r ´ deg i p x q edges of this type colored i incident with x for each i P r k s . Notethat after this coloring,(7) deg i p x q “ r for each x P V. For i P r k s , let a i , b i , c i , d i be the number of edges colored i of the form W, W
Y t u u , W Yt u u , W Y t u u where W Ď V , respectively. We color the edges of the form t u u sothat there are exactly e i : “ rn { ´ rm ` a i ` b i ` c i edges of this type colored i for i P r k s . Since 4 | rn , and n ą m , e i is a positive integerfor i P r k s . We claim that all edges of the form t u u will be colored, or equivalently, ř ki “ e i “ ` n ´ m ˘ . k ÿ i “ e i “ k ÿ i “ p rn ´ rm ` a i ` b i ` c i q “ rkn ´ rkm ` k ÿ i “ a i ` k ÿ i “ b i ` k ÿ i “ c i “ n ˆ n ´ ˙ ´ m ˆ n ´ ˙ ` ˆ m ˙ ` p n ´ m q ˆ m ˙ ` ˆ n ´ m ˙ˆ m ˙ “ ˆ n ˙ ´ m ˆ n ´ ˙ ` ˆ m ˙ ` p n ´ m q ˆ m ˙ ` ˆ n ´ m ˙ˆ m ˙ p q , p q “ ˆ n ´ m ˙ . To complete the proof, we show that deg i p u q “ r p n ´ m q for i P r k s . First note thatfor i P r k s , rm “ ř x P V deg i p x q “ a i ` b i ` c i ` d i . Therefore,deg i p u q “ b i ` c i ` d i ` e i “ p a i ` b i ` c i ` d i ` e i q ´ p a i ` b i ` c i ` d i q“ rn ´ rm “ r p n ´ m q . Combining this with (7) implies that (6) is satisfied, and the proof is complete. (cid:3) h “ Theorem 5.1.
For n ě . m , any partial r -factorization of K m can be extendedto an r -factorization of K n if and only if | rn and r | ` n ´ ˘ . XTENDING EDGE-COLORINGS OF COMPLETE HYPERGRAPHS 11
Proof.
The necessity is obvious. To prove the sufficiency, we need to show that theedges of F : “ Ą K m can be colored with k : “ ` n ´ ˘ { r colors so that (6) is satisfied.First we color the edges of the form W Y t u u where W Ď V and | W | “
4. We colorthese edges greedily so that deg i p x q ď r for each x P V and i P r k s . We claim that thiscoloring can be done in such a way that all edges of this type are colored. Suppose bycontrary that there is an edge of the form t x, y, z, w, u u with x, y, z, w P V that cannot be colored. This implies that for each i P r k s either deg i p x q “ r or deg i p y q “ r ordeg i p z q “ r or deg i p w q “ r . Thus for each i P r k s , deg i p x q` deg i p y q` deg i p z q` deg i p w q ě r . On the one hand, ř ki “ ` deg i p x q ` deg i p y q ` deg i p z q ` deg i p w q ˘ ě rk “ ` n ´ ˘ , andon the other hand, ř ki “ ` deg i p x q ` deg i p y q ` deg i p z q ` deg i p w q ˘ ď r ` m ´ ˘ ` p n ´ m q ` m ´ ˘ ´ s . Thus, we have4 r ˆ m ´ ˙ ` p n ´ m q ˆ m ´ ˙ ´ s ě ˆ n ´ ˙ . which is equivalent to g p n, m q : “ n ´ n ` n ´ m n ` m n ´ mn ` n ` m ´ m ` m ` m ` ď n ą m and m ě
6, we have g p n, m q : “ n ´ n p n ´ q ´ m ` m ` p n ´ m q ` ¯ ` m ´ m p m ´ q ` m ` ¯ ` ą m p m ´ q ´ m ` m “ m ` m ą , which is a contradiction, and therefore, all edges in F of the form W Y t u u where W Ď V and | W | “ W Y t u u where W Ď V and | W | “ i p x q ď r for each x P V and i P r k s . We show that this is possible. If bycontrary, some edge t x, y, z, u u with x, y, z P V remains uncolored, then for each i P r k s either deg i p x q “ r or deg i p y q “ r or deg i p z q “ r , and so deg i p x q ` deg i p y q ` deg i p z q ě r .We have ˆ n ´ ˙ “ rk ď k ÿ i “ ` deg i p x q ` deg i p y q ` deg i p z q ˘ ď r ˆ m ´ ˙ ` p n ´ m q ˆ m ´ ˙ ` ˆ m ´ ˙ˆ n ´ m ˙ ´ s . which is equivalent to g p n, m q : “ n ´ n ´ m n ` mn ´ n ` m n ´ m n ´ mn ` n ´ m ´ m ` m ` m ` ď
0. We show that since n ą m and m ě
6, we have g p n, m q ą
0, which is a contradiction, and therefore, all edges in F ofthe form W Y t u u where W Ď V and | W | “ XTENDING EDGE-COLORINGS OF COMPLETE HYPERGRAPHS 12
First, note that for m ě m ´ m ´ m ` ą
0. Therefore, g p n, m q “ n ´ n p n ´ q ´ m ` m ´ ¯ ` n ´ m ´ m ´ m ` ¯ ´ p m ` m ´ m ´ m ´ qą m ´ m p m ´ q ´ m ` m ´ ¯ ´ p m ` m ´ m ´ m ´ q“ m ´ m ` m ` m ` ą . Now we greedily color all the edges of the form W Y t u u where W Ď V and | W | “ i p x q ď r for each x P V and i P r k s . We show that this is possible. If bycontrary, some edge t x, y, u u with x, y P V remains uncolored, then for each i P r k s either deg i p x q “ r or deg i p y q “ r , and so deg i p x q ` deg i p y q ě r . We have ˆ n ´ ˙ ď k ÿ i “ ` deg i p x q ` deg i p y q ˘ ď r ˆ m ´ ˙ ` p n ´ m q ˆ m ´ ˙ ` ˆ m ´ ˙ˆ n ´ m ˙ ` p m ´ q ˆ n ´ m ˙ ´ s . Using Mathematica it can be shown that this inequality does not have any real solutionunder the constraints m ě , n ě . m . Therefore, all edges of the form W Y t u u where W Ď V and | W | “
2, can be colored.Since for each x P V , k ÿ i “ ` r ´ deg i p x q ˘ “ ˆ n ´ ˙ ´ ˆ m ´ ˙ ´ p n ´ m q ˆ m ´ ˙ ´ ˆ m ´ ˙ˆ n ´ m ˙ ´ p m ´ q ˆ n ´ m ˙ “ ˆ n ´ m ˙ , we can color all the edges of the form t w, u u where w P V so that for each x P V ,there are r ´ deg i p x q edges of this type colored i incident with x for each i P r k s .For i P r k s , let a i , b i , c i , d i , e i be the number of edges colored i of the form W, W Yt u u , W Y t u u , W Y t u u , W Y t u u where W Ď V , respectively. We color the edges ofthe form t u u so that there exactly f i : “ rn { ´ rm ` a i ` b i ` c i ` d i edges of this type colored i for i P r k s . Since 5 | rn , and n ě . m ą m , e i is apositive integer for i P r k s . We claim that all edges of the form t u u will be colored, orequivalently, ř ki “ f i “ ` n ´ m ˘ . XTENDING EDGE-COLORINGS OF COMPLETE HYPERGRAPHS 13 k ÿ i “ f i “ k ÿ i “ p rn ´ rm ` a i ` b i ` c i ` d i q“ rkn ´ rkm ` k ÿ i “ a i ` k ÿ i “ b i ` k ÿ i “ c i ` k ÿ i “ d i “ ˆ n ˙ ´ m ˆ n ´ ˙ ` ˆ m ˙ ` p n ´ m q ˆ m ˙ ` ˆ n ´ m ˙ˆ m ˙ ` ˆ n ´ m ˙ˆ m ˙ “ ˆ n ´ m ˙ . To complete the proof, we show that deg i p u q “ r p n ´ m q for i P r k s . First note thatfor i P r k s , rm “ ř x P V deg i p x q “ a i ` b i ` c i ` d i ` e i . Therefore,deg i p u q “ b i ` c i ` d i ` e i ` f i “ p a i ` b i ` c i ` d i ` e i ` f i q ´ p a i ` b i ` c i ` d i ` e i q“ rn ´ rm “ r p n ´ m q . (cid:3) Concluding Remarks and Open Problems (1) At this point, it is not clear to use how to extend the results of Sections 4 and5 without dealing with heavy computation. We believe that for n ě hm , anypartial r -factorization of K hm can be extended to an r -factorization of K hn if andonly if the obvious necessary divisibility conditions are satisfied.(2) To embed a partial r -factorization of K n z K hm into an r -factorization of K hn , webelieve that the conditions (N1)–(N4) of Lemma 3.2 are sufficient, but we donot know how to go beyond Corollary 3.3.(3) A partial r -factorization S Ď K hn is critical if it can be extended to exactly one r -factorization of K hn , but removal of any element of S destroys the uniquenessof the extension, and | S | is the size of the critical partial r -factorization. It isdesirable to find good bounds for the smallest and largest sizes of critical partial r -factorizations.(4) Another interesting problem is finding conditions under which a partial r -factorization of S Ď ` r n s h ˘ can be extended to a cyclic r -factorization of ` r n s h ˘ . Acknowledgement
The author’s research is supported by Summer Faculty Fellowship at ISU, and NSAGrant H98230-16-1-0304. The author wishes to thank Lana Kuhle, Dan Roberts andthe anonymous referees for their constructive feedback on the first draft of this paper.
XTENDING EDGE-COLORINGS OF COMPLETE HYPERGRAPHS 14
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