Extending regular edge-colorings of complete hypergraphs

A coloring (partition) of the collection $(\genfrac{}{}{0.0pt}{}{X}{h})$ 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 $(\genfrac{}{}{0.0pt}{}{X}{h})$ is extendible to an $s$-regular coloring of $(\genfrac{}{}{0.0pt}{}{Y}{h})$ for $s⩾r$ and $Y⊋X$. The case $\text{◂,▸}h=2,r=s=1$ 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 $h=4,\text{◂⩾▸}|Y|⩾4|X|$ and $h=5,\text{◂⩾▸}|Y|⩾5|X|$.     

Almost regular edge colorings and regular decompositions of complete graphs

For k = 1 and k = 2, we prove that the obvious necessary numerical conditions for packing t pairwise edge‐disjoint k‐regular subgraphs of specified orders m₁,m₂,… ,m_t in the complete graph of order n are also sufficient. To do so, we present an edge‐coloring technique which also yields new proofs of various known results on graph factorizations. For example, a new construction for Hamilton cycle decompositions of complete graphs is given. © 2008 Wiley Periodicals, Inc. J Combin Designs 16: 499–506, 2008     

Compact cyclic edge-colorings of graphs

The paper is devoted to a model of compact cyclic edge-coloring of graphs. This variant of edge-coloring finds its applications in modeling schedules in production systems, in which production proceeds in a cyclic way. We point out optimal colorings for some graph classes and we construct graphs which cannot be colored in a compact cyclic manner. Moreover, we prove some theoretical properties of considered coloring model such as upper bounds on the number of colors in optimal compact cyclic coloring.     

On splittable colorings of graphs and hypergraphs

The notion of a split coloring of a complete graph was introduced by Erd?s and Gyárfás [ 7 ] as a generalization of split graphs. In this work, we offer an alternate interpretation by comparing such a coloring to the classical Ramsey coloring problem via a two‐round game played against an adversary. We show that the techniques used and bounds obtained on the extremal (r,m)‐split coloring problem of [ 7 ] are closer in nature to the Turán theory of graphs rather than Ramsey theory. We extend the notion of these colorings to hypergraphs and provide bounds and some exact results. © 2002 Wiley Periodicals, Inc. J Graph Theory 40: 226–237, 2002     

Covering complete partite hypergraphs by monochromatic components

A well-known special case of a conjecture attributed to Ryser (actually appeared in the thesis of Henderson (1971)) states that $k$-partite intersecting hypergraphs have transversals of at most $k?1$ vertices. An equivalent form of the conjecture in terms of coloring of complete graphs is formulated in Gyárfás (1977): if the edges of a complete graph $K$ are colored with $k$ colors then the vertex set of $K$ can be covered by at most $k?1$ sets, each forming a connected graph in some color. It turned out that the analogue of the conjecture for hypergraphs can be answered: it was proved in Király (2013) that in every $k$-coloring of the edges of the $r$-uniform complete hypergraph $K^r$ ($r\ge 3$), the vertex set of $K^r$ can be covered by at most $?k∕r?$ sets, each forming a connected hypergraph in some color.Here we investigate the analogue problem for complete $r$-uniform $r$-partite hypergraphs. An edge coloring of a hypergraph is called spanning if every vertex is incident to edges of every color used in the coloring. We propose the following analogue of Ryser’s conjecture.In every spanning $(r+t)$-coloring of the edges of a complete $r$-uniform $r$-partite hypergraph, the vertex set can be covered by at most $t+1$ sets, each forming a connected hypergraph in some color.We show that the conjecture (if true) is best possible. Our main result is that the conjecture is true for $1\le t\le r?1$. We also prove a slightly weaker result for $t\ge r$, namely that $t+2$ sets, each forming a connected hypergraph in some color, are enough to cover the vertex set.To build a bridge between complete $r$-uniform and complete $r$-uniform $r$-partite hypergraphs, we introduce a new notion. A hypergraph is complete $r$-uniform $(r,?)$-partite if it has all $r$-sets that intersect each partite class in at most $?$ vertices (where $1\le ?\le r$).Extending our results achieved for $?=1$, we prove that for any $r\ge 3,2\le ?\le r,k\ge 1+r??$, in every spanning $k$-coloring of the edges of a complete $r$-uniform $(r,?)$-partite hypergraph, the vertex set can be covered by at most $1+?\frac{k?r+??1}{?}?$ sets, each forming a connected hypergraph in some color.     

Large monochromatic components in colorings of complete 3-uniform hypergraphs

Let f(n,r) be the largest integer m with the following property: if the edges of the complete 3-uniform hypergraph are colored with r colors then there is a monochromatic component with at least m vertices. Here we show that and . Both results are sharp under suitable divisibility conditions (namely if n is divisible by 7, or by 6 respectively).     

图的区间边着色的收缩图方法

图G的一个用了颜色1,2,…,t的边着色称为区间t-着色,如果所有t种颜色都被用到,并且关联于G的同一个顶点的边上的颜色是各不相同的,且这些颜色构成了一个连续的整数区间.G称作是可区间着色的,如果对某个正整数t,G有一个区间t-着色.所有可区间着色的图构成的集合记作■.对图G∈■,使得G有一个区间t-着色的t的最小值和最大值分别记作ω(G)和W(G).现给出了图的区间着色的收缩图方法.利用此方法,我们对双圈图G∈■,证明了ω(G)=△(G)或△(G)+1,并且完全确定了ω(G)=△(G)及ω(G)=△(G)+1的双圈图类.     

List colorings of multipartite hypergraphs

Let χ_l(G) denote the list chromatic number of the r‐uniform hypergraph G. Extending a result of Alon for graphs, Saxton and the second author used the method of containers to prove that, if G is simple and d‐regular, then . To see how close this inequality is to best possible, we examine χ_l(G) when G is a random r‐partite hypergraph with n vertices in each class. The value when r = 2 was determined by Alon and Krivelevich; here we show that almost surely, where d is the expected average degree of G and . The function g(r,α) is defined in terms of “preference orders” and can be determined fairly explicitly. This is enough to show that the container method gives an optimal lower bound on χ_l(G) for r = 2 and r = 3, but, perhaps surprisingly, apparently not for r ≥ 4.     

Cube factorizations of complete graphs

A cube factorization of the complete graph on n vertices, K_n, is a 3‐factorization of K_n in which the components of each factor are cubes. We show that there exists a cube factorization of K_n if and only if n ≡ 16 (mod 24), thus providing a new family of uniform 3‐factorizations as well as a partial solution to an open problem posed by Kotzig in 1979. © 2004 Wiley Periodicals, Inc.     

Decomposition of complete uniform multi‐hypergraphs into Berge paths and cycles

In 2015, Bryant, Horsley, Maenhaut, and Smith, generalizing a well‐known conjecture by Alspach, obtained the necessary and sufficient conditions for the decomposition of the complete multigraph into cycles of arbitrary lengths, where I is empty, when is even and I is a perfect matching, when is odd. Moreover, Bryant in 2010, verifying a conjecture by Tarsi, proved that the obvious necessary conditions for packing pairwise edge‐disjoint paths of arbitrary lengths in are also sufficient. In this article, first, we obtain the necessary and sufficient conditions for packing edge‐disjoint cycles of arbitrary lengths in . Then, applying this result, we investigate the analogous problem of the decomposition of the complete uniform multihypergraph into Berge cycles and paths of arbitrary given lengths. In particular, we show that for every integer , and , can be decomposed into Berge cycles and paths of arbitrary lengths, provided that the obvious necessary conditions hold, thereby generalizing a result by Kühn and Osthus on the decomposition of into Hamilton Berge cycles.     

Hamilton cycle rich two‐factorizations of complete graphs

For all integers n ≥ 5, it is shown that the graph obtained from the n‐cycle by joining vertices at distance 2 has a 2‐factorization is which one 2‐factor is a Hamilton cycle, and the other is isomorphic to any given 2‐regular graph of order n. This result is used to prove several results on 2‐factorizations of the complete graph K_n of order n. For example, it is shown that for all odd n ≥ 11, K_n has a 2‐factorization in which three of the 2‐factors are isomorphic to any three given 2‐regular graphs of order n, and the remaining 2‐factors are Hamilton cycles. For any two given 2‐regular graphs of even order n, the corresponding result is proved for the graph K_n ‐ I obtained from the complete graph by removing the edges of a 1‐factor. © 2004 Wiley Periodicals, Inc.     

Edge‐transitive homogeneous factorizations of complete uniform hypergraphs

For a finite set V and a positive integer k with , letting be the set of all k‐subsets of V, the pair is called the complete k‐hypergraph on V, while each k‐subset of V is called an edge. A factorization of the complete k‐hypergraph of index , simply a ‐factorization of order n, is a partition of the edges into s disjoint subsets such that each k‐hypergraph , called a factor, is a spanning subhypergraph of . Such a factorization is homogeneous if there exist two transitive subgroups G and M of the symmetric group of degree n such that G induces a transitive action on the set and M lies in the kernel of this action. In this article, we give a classification of homogeneous factorizations of that admit a group acting transitively on the edges of . It is shown that, for and , there exists an edge‐transitive homogeneous ‐factorization of order n if and only if is one of (32, 3, 5), (32, 3, 31), (33, 4, 5), , and , where and q is a prime power with .     

Colorings versus list colorings of uniform hypergraphs

Let $r$ be an integer with $r\ge 2$ and $G$ be a connected $r$-uniform hypergraph with $m$ edges. By refining the broken cycle theorem for hypergraphs, we show that if $k>\text{◂/▸}\frac{m-1}{\ln (1+\sqrt{2})}\approx \text{◂⋅▸}1.135\left(m-1\right)$, then the $k$-list assignment of $G$ admitting the fewest colorings is the constant list assignment. This extends the previous results of Donner, Thomassen, and the current authors for graphs.     

On tight 6-cycle decompositions of complete 3-uniform hypergraphs

The complete 3-uniform hypergraph of order v has a vertex set V of size v and the set of all 3-element subsets of V as its edge set. A tight 6-cycle is a hypergraph with vertex set $\{a,b,c,d,e,f\}$ and edge set $\{\{a,b,c\},\{b,c,d\},\{c,d,e\},\{d,e,f\},\{e,f,a\},\{f,a,b\}\}$. We show that there exists a decomposition of the complete 3-uniform hypergraph of order v into isomorphic copies of a tight 6-cycle if and only if $v\equiv 1$, 2, 10, 20, 28, or $29(\mathrm{mod}36)$.     

A note on embedding into product spaces

Using factorization properties of an operator ideal over a Banach space, it is shown how to embed a locally convex space from the corresponding Grothendieck space ideal into a suitable power of E, thus achieving a unified treatment of several embedding theorems involving certain classes of locally convex spaces.     

A precolouring extension of Vizing's theorem

Fix a palette of colors, a graph with maximum degree , and a subset of the edge set with minimum distance between edges at least . If the edges of are arbitrarily precoloured from , then there is guaranteed to be a proper edge-coloring using only colors from that extends the precolouring on to the entire graph. This result is a first general precolouring extension form of Vizing's theorem, and it proves a conjecture of Albertson and Moore under a slightly stronger distance requirement. We also show that the condition on the distance can be lowered to when the graph contains no cycle of length .     

加权Orlicz-Lorentz鞅空间的原子分解及其应用

本文研究加权Orlicz-Lorentz鞅空间的原子分解理论.利用原子分解方法,获得了加权Orlicz-Lorentz鞅空间的内插理论以及鞅变换算子的有界性证明.上述结果推广了Orlicz和Lorentz鞅空间理论相关结果.     

Decomposing complete edge-chromatic graphs and hypergraphs. Revisited

A d-graph is a complete graph whose edges are colored by d colors, that is, partitioned into d subsets some of which might be empty. We say that a d-graph is complementary connected (CC) if the complement to each chromatic component of is connected on V. We prove that every such d-graph contains a sub-d-graph Π or , where Π has four vertices and two non-empty chromatic components each of which is a P₄, while is a three-colored triangle. This statement implies that each Π- and -free d-graph is uniquely decomposable in accordance with a tree whose leaves are the vertices of V and the interior vertices of T are labeled by the colors 1,…d. Such a tree is naturally interpreted as a positional game form (with perfect information and without moves of chance) of d players I={1,…,d} and n outcomes V={v₁,…,v_n}. Thus, we get a one-to-one correspondence between these game forms and Π- and -free d-graphs. As a corollary, we obtain a characterization of the normal forms of positional games with perfect information and, in case d=2, several characterizations of the read-once Boolean functions. These results are not new; in fact, they are 30 and, in case d=2, even 40 years old. Yet, some important proofs did not appear in English.Gyárfás and Simonyi recently proved a similar decomposition theorem for the -free d-graphs. They showed that each -free d-graph can be obtained from the d-graphs with only two non-empty chromatic components by successive substitutions. This theorem is based on results by Gallai, Lovász, Cameron and Edmonds. We obtain some new applications of these results.