Color-bounded hypergraphs, IV: Stable colorings of hypertrees

We consider vertex colorings of hypergraphs in which lower and upper bounds are prescribed for the largest cardinality of a monochromatic subset and/or of a polychromatic subset in each edge. One of the results states that for any integers s≥2 and a≥2 there exists an integer f(s,a) with the following property. If an interval hypergraph admits some coloring such that in each edge E_i at least a prescribed number s_i≤s of colors occur and also each E_i contains a monochromatic subset with a prescribed number a_i≤a of vertices, then a coloring with these properties exists with at most f(s,a) colors. Further results deal with estimates on the minimum and maximum possible numbers of colors and the time complexity of determining those numbers or testing colorability, for various combinations of the four color bounds prescribed. Many interesting problems remain open.     

Color-bounded hypergraphs, I: General results

The concept of color-bounded hypergraph is introduced here. It is a hypergraph (set system) with vertex set X and edge set E={E₁,…,E_m}, where each edge E_i is associated with two integers s_i and t_i such that 1≤s_i≤t_i≤|E_i|. A vertex coloring φ:X→N is considered to be feasible if the number of colors occurring in E_i satisfies s_i≤|φ(E_i)|≤t_i, for all i≤m.Color-bounded hypergraphs generalize the concept of ‘mixed hypergraphs’ introduced by Voloshin [V. Voloshin, The mixed hypergraphs, Computer Science Journal of Moldova 1 (1993) 45-52], and a recent model studied by Drgas-Burchardt and ?azuka [E. Drgas-Burchardt, E. ?azuka, On chromatic polynomials of hypergraphs, Applied Mathematics Letters 20 (12) (2007) 1250-1254] where only lower bounds s_i were considered.We discuss the similarities and differences between our general model and the more particular earlier ones. An important issue is the chromatic spectrum-strongly related to the chromatic polynomial-which is the sequence whose kth element is the number of allowed colorings with precisely k colors (disregarding color permutations). Problems concerning algorithmic complexity are also considered.     

Color-bounded hypergraphs, II: Interval hypergraphs and hypertrees

A color-bounded hypergraph is a hypergraph (set system) with vertex set X and edge set E={E₁,…,E_m}, together with integers s_i and t_i (1≤s_i≤t_i≤|E_i|) for i=1,…,m. A vertex coloring φ is feasible if the number of colors occurring in edge E_i satisfies s_i≤|φ(E_i)|≤t_i, for every i≤m.In this paper we point out that hypertrees-hypergraphs admitting a representation over a (graph) tree where each hyperedge E_i induces a subtree of the underlying tree-play a central role concerning the set of possible numbers of colors that can occur in feasible colorings. We also consider interval hypergraphs and circular hypergraphs, where the underlying graph is a path or a cycle, respectively. Sufficient conditions are given for a ‘gap-free’ chromatic spectrum; i.e., when each number of colors is feasible between minimum and maximum. The algorithmic complexity of colorability is studied, too.Compared with the ‘mixed hypergraphs’-where ‘D-edge’ means (s_i,t_i)=(2,|E_i|), while ‘C-edge’ assumes (s_i,t_i)=(1,|E_i|−1)-the differences are rather significant.     

Graphs of Large Girth with Prescribed Partial Circular Colourings

This paper completes the constructive proof of the following result: Suppose p/q2 is a rational number, A is a finite set and f₁,f₂,···,f_n are mappings from A to {0,1,···,p–1}. Then for any integer g, there is a graph G=(V,E) of girth at least g with such that G has exactly n (p,q)-colourings (up to equivalence) g₁,g₂,···,g_n, and each g_i is an extension of f_i. A probabilistic proof of this result was given in [8]. A constructive proof of the case p/q3 was given in [7].This research was partially supported by the National Science Council under grant NSC91-2115-M-110-004     

Quasi-random hypergraphs

We introduce an equivalence class of varied properties for hypergraphs. Any hypergraph possessing any one of these properties must of necessity possess them all. Since almost all random hypergraphs share these properties, we term these properties quasi-random. With these results, it becomes quite easy to show that many natural explicit constructions result in hypergraphs which imitate random hypergraphs in a variety of ways.     

Construction of uniquely H‐colorable graphs

We shall prove that for any graph H that is a core, if χ(G) is large enough, then H × G is uniquely H‐colorable. We also give a new construction of triangle free graphs, which are uniquely n‐colorable. © 1999 John Wiley & Sons, Inc. J Graph Theory 30: 1–6, 1999     

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.     

Oriented hypergraphs: Balanceability

An oriented hypergraph is an oriented incidence structure that extends the concepts of signed graphs, balanced hypergraphs, and balanced matrices. We introduce hypergraphic structures and techniques that generalize the circuit classification of the signed graphic frame matroid to any oriented hypergraphic incidence matrix via its locally-signed-graphic substructure. To achieve this, Camion's algorithm is applied to oriented hypergraphs to provide a generalization of reorientation sets and frustration that is only well-defined on balanceable oriented hypergraphs. A simple partial characterization of unbalanceable circuits extends the applications to representable matroids demonstrating that the difference between the Fano and non-Fano matroids is one of balance.     

r-tuple colorings of uniquely colorable graphs

An r-tuple coloring of a graph is one in which r colors are assigned to each point of the graph so that the sets of colors assigned to adjacent points are always disjoint. We investigate the question of whether a uniquely n-colorable graph can receive an r-tuple coloring with fewer than nr colors. We show that this cannot happen for n=3 and r=2 and that for a given n and r to establish the conjecture that no uniquely n-colorable graph can receive an r-tuple coloring from fewer than nr colors it suffices to prove it for on a finite set of uniquely n-colorable graphs.     

C‐perfect hypergraphs

Let ${\mathcal{H}}=({{X}},{\mathcal{E}})Let ${\mathcal{H}}=({{X}},{\mathcal{E}})$ be a hypergraph with vertex set X and edge set ${\mathcal{E}}$. A C‐coloring of ${\mathcal{H}}$ is a mapping ?:X→? such that |?(E)|<|E| holds for all edges ${{E}}\in{\mathcal{E}}$ (i.e. no edge is multicolored). We denote by $\bar{\chi}({\mathcal{H}})$ the maximum number |?(X)| of colors in a C‐coloring. Let further $\alpha({\mathcal{H}})$ denote the largest cardinality of a vertex set S?X that contains no ${{E}}\in{\mathcal{E}}$, and $\tau({\mathcal{H}})=|{{X}}|-\alpha({\mathcal{H}})$ the minimum cardinality of a vertex set meeting all $E \in {\mathcal{E}}$. The hypergraph ${\mathcal{H}}$ is called C‐perfect if $\bar{\chi}({\mathcal{H}}\prime)=\alpha({\mathcal{H}}\prime)$ holds for every induced subhypergraph ${\mathcal{H}}\prime\subseteq{\mathcal{H}}$. If ${\mathcal{H}}$ is not C‐perfect but all of its proper induced subhypergraphs are, then we say that it is minimally C‐imperfect. We prove that for all r, k∈? there exists a finite upper bound h(r, k) on the number of minimally C‐imperfect hypergraphs ${\mathcal{H}}$ with $\tau({\mathcal{H}})\le {{k}}$ and without edges of more than r vertices. We give a characterization of minimally C‐imperfect hypergraphs that have τ=2, which also characterizes implicitly the C‐perfect ones with τ=2. From this result we derive an infinite family of new constructions that are minimally C‐imperfect. A characterization of minimally C‐imperfect circular hypergraphs is presented, too. © 2009 Wiley Periodicals, Inc. J Graph Theory 64: 132–149, 2010     

Judicious bisection of hypergraphs

Judicious bisection of hypergraphs asks for a balanced bipartition of the vertex set that optimizes several quantities simultaneously.In this paper,we prove that if G is a hypergraph with n vertices and m_i edges of size i for i=1,2,...,k,then G admits a bisection in which each vertex class spans at most(m_1)/2+1/4m_2+…+(1/(2~k)+m_k+o(m_1+…+m_k) edges,where G is dense enough or △(G)= o(n) but has no isolated vertex,which turns out to be a bisection version of a conjecture proposed by Bollobas and Scott.     

Counting acyclic hypergraphs

Acyclic hypergraphs are analogues of forests in graphs. They are very useful in the design of databases. The number of distinct acyclic uniform hypergraphs withn labeled vertices is studied. With the aid of the principle of inclusion-exclusion, two formulas are presented. One is the explicitformula for strict (d)-connected acyclic hypergraphs, the other is the recurrence formula for linear acyclic hypergraphs.     

The Cartesian product of hypergraphs

We show that every simple, (weakly) connected, possibly directed and infinite, hypergraph has a unique prime factor decomposition with respect to the (weak) Cartesian product, even if it has infinitely many factors. This generalizes previous results for graphs and undirected hypergraphs to directed and infinite hypergraphs. The proof adopts the strategy outlined by Imrich and ?erovnik for the case of graphs and introduces the notion of diagonal‐free grids as a replacement of the chord‐free 4‐cycles that play a crucial role in the case of graphs. This leads to a generalization of relation Δ on the arc set, whose convex hull is shown to coincide with the product relation of the prime factorization. © 2011 Wiley Periodicals, Inc. J Graph Theory     

Hamiltonian chains in hypergraphs

A cyclic ordering of the vertices of a k‐uniform hypergraph is called a hamiltonian chain if any k consecutive vertices in the ordering form an edge. For k = 2 this is the same as a hamiltonian cycle. We consider several natural questions about the new notion. The main result is a Dirac‐type theorem that provides a sufficient condition for finding hamiltonian chains in k‐uniform hypergraphs with large (k − 1)‐minimal degree. If it is more than than the hypergraph contains a hamiltonian chain. © 1999 Wiley & Sons, Inc. J Graph Theory 30: 205–212, 1999     

5-一致超图的全横贯

设H=(V,E)是以V为顶点集, E为(超)边集的超图. 如果H的每条边均含有k个顶点, 则称H是k-一致超图. 超图H的点子集T称为它的一个横贯, 如果T 与H 的每条边均相交. 超图H的全横贯是指它的一个横贯T, 并且T还满足如下性质: T中每个顶点均至少有一个邻点在T中. H 的全横贯数定义为H 的最小全横贯所含顶点的数目, 记作\tau_{t}(H). 对于整数k\geq 2, 令b_{k}=\sup_{H\in{\mathscr{H}}_{k}}\frac{\tau_{t}(H)}{n_{H}+m_{H}}, 其中n_H=|V|, m_H=|E|, {\mathscr{H}}_{k} 表示无孤立点和孤立边以及多重边的k-一致超图类. 最近, Bujt\'as和Henning等证明了如下结果: b_{2}=\frac{2}{5}, b_{3}=\frac{1}{3}, b_{4}=\frac{2}{7}; 当k\geq 5 时, 有b_{k}\leq \frac{2}{7}以及b_{6}\leq \frac{1}{4}; 当k\geq 7 时, b_{k}\leq \frac{2}{9}. 证明了对5-一致超图, b_{5}\leq \frac{4}{15}, 从而改进了当k=5 时b_k的上界.     

Exploring hypergraphs with martingales

Recently, in [Random Struct Algorithm 41 (2012), 441–450] we adapted exploration and martingale arguments of Nachmias and Peres [ALEA Lat Am J Probab Math Stat 3 (2007), 133–142], in turn based on ideas of Martin‐Löf [J Appl Probab 23 (1986), 265–282], Karp [Random Struct Alg 1 (1990), 73–93] and Aldous [Ann Probab 25 (1997), 812–854], to prove asymptotic normality of the number L₁ of vertices in the largest component of the random r‐uniform hypergraph in the supercritical regime. In this paper we take these arguments further to prove two new results: strong tail bounds on the distribution of L₁, and joint asymptotic normality of L₁ and the number M₁ of edges of in the sparsely supercritical case. These results are used in [Combin Probab Comput 25 (2016), 21–75], where we enumerate sparsely connected hypergraphs asymptotically. © 2016 Wiley Periodicals, Inc. Random Struct. Alg., 50, 325–352, 2017     

Greedy colorings of uniform hypergraphs

We give a very short proof of an Erd?s conjecture that the number of edges in a non‐2‐colorable n‐uniform hypergraph is at least f(n)2ⁿ, where f(n) goes to infinity. Originally it was solved by József Beck in 1977, showing that f(n) at least clog n. With an ingenious recoloring idea he later proved that f(n) ≥ cn^1/3+o(1). Here we prove a weaker bound on f(n), namely f(n) ≥ cn^1/4. Instead of recoloring a random coloring, we take the ground set in random order and use a greedy algorithm to color. The same technique works for getting bounds on k‐colorability. It is also possible to combine this idea with the Lovász Local Lemma, reproving some known results for sparse hypergraphs (e.g., the n‐uniform, n‐regular hypergraphs are 2‐colorable if n ≥ 8). © 2009 Wiley Periodicals, Inc. Random Struct. Alg., 2009