Lines in hypergraphs

One of the De Bruijn-Erd?s theorems deals with finite hypergraphs where every two vertices belong to precisely one hyperedge. It asserts that, except in the perverse case where a single hyperedge equals the whole vertex set, the number of hyperedges is at least the number of vertices and the two numbers are equal if and only if the hypergraph belongs to one of simply described families, near-pencils and finite projective planes. Chen and Chvátal proposed to define the line uv in a 3-uniform hypergraph as the set of vertices that consists of u, v, and all w such that {u;v;w} is a hyperedge. With this definition, the De Bruijn-Erd?s theorem is easily seen to be equivalent to the following statement: If no four vertices in a 3-uniform hypergraph carry two or three hyperedges, then, except in the perverse case where one of the lines equals the whole vertex set, the number of lines is at least the number of vertices and the two numbers are equal if and only if the hypergraph belongs to one of two simply described families. Our main result generalizes this statement by allowing any four vertices to carry three hyperedges (but keeping two forbidden): the conclusion remains the same except that a third simply described family, complements of Steiner triple systems, appears in the extremal case.     

On the Heterochromatic Number of Hypergraphs Associated to Geometric Graphs and to Matroids

The heterochromatic number h _c (H) of a non-empty hypergraph H is the smallest integer k such that for every colouring of the vertices of H with exactly k colours, there is a hyperedge of H all of whose vertices have different colours. We denote by ν(H) the number of vertices of H and by τ(H) the size of the smallest set containing at least two vertices of each hyperedge of H. For a complete geometric graph G with n ≥ 3 vertices let H = H(G) be the hypergraph whose vertices are the edges of G and whose hyperedges are the edge sets of plane spanning trees of G. We prove that if G has at most one interior vertex, then h _c (H) = ν(H) ? τ(H) + 2. We also show that h _c (H) = ν(H) ? τ(H) + 2 whenever H is a hypergraph with vertex set and hyperedge set given by the ground set and the bases of a matroid, respectively.     

Independent trees in graphs

IfG is a finite undirected graph ands is a vertex ofG, then two spanning treesT ₁ andT ₂ inG are calleds — independent if for each vertexx inG the paths fromx tos inT ₁ andT ₂ are openly disjoint. It is known that the following statement is true fork3: IfG isk-connected, then there arek pairwises — independent spanning, trees inG. As a main result we show that this statement is also true fork=4 if we restrict ourselves to planar graphs. Moreover we consider similar statements for weaklys — independent spanning trees (i.e., the tree paths from a vertex tos are edge disjoint) and for directed graphs.     

Covers in hypergraphs

Let α(H) denote the stability number of a hypergraphH. The covering number ?(H) is defined as the minimal number of edges fromH to cover its vertex setV(H). The main result is the following extension of König’s wellknown theorem: If α(H′)≧|V(H′)|/2 holds for every section hypergraphH′ ofH then ?(H)≦α(H). This theorem is applied to obtain upper bounds on certain covering numbers of graphs and hypergraphs. In par ticular, we prove a conjecture of B. Bollobás involving the hypergraph Turán numbers.     

Finding Scores in Tournaments

A tournamentT_nis an orientation of the complete graph onnvertices. We continue the algorithmic study initiated by10of recognizing various directed trees in tournaments. Hell and Rosenfeld studied the complexity of finding various oriented paths in tournaments by probing edge directions. Here, we investigate the complexity of finding a vertex of prescribed outdegree (or indegree) in the same model. We show that the complexity of finding a vertex of outdegreek( ≤ (n − 1)/2) inT_nis Θ(nk). This bound is in sharp contrast to the Θ(n) bound for selection in the case of transitive tournaments. We also establish tight bounds for finding vertices of prescribed degree from the adjacency matrix of general directed/undirected graphs. These bounds generalize the classical bound of11for finding a sink (a vertex of outdegree 0 and indegreen − 1) in a directed graph.     

Transversal numbers of uniform hypergraphs

The transversal number(H) of a hypergraphH is the minimum cardinality of a set of vertices that intersects all edges ofH. Fork 1 definec _k =sup (H)/(m + n), whereH ranges over allk-uniform hypergraphs withn vertices andm edges. Applying probabilistic arguments we show thatc _k = (1 +o(1))log _e k/k. This settles a problem of Tuza.Research supported in part by Allon Fellowship and by a grant from the Bat Sheva de Rothschild Foundation.     

Reconstructing graphs as subsumed graphs of hypergraphs,and some self-complementary triple systems

LetV be a set ofn elements. The set of allk-subsets ofV is denoted . Ak-hypergraph G consists of avertex-set V(G) and anedgeset , wherek≥2. IfG is a 3-hypergraph, then the set of edges containing a given vertexvεV(G) define a graphG _v . The graphs {G _v νvεV(G)} aresubsumed byG. Each subsumed graphG _v is a graph with vertex-setV(G) − v. They can form the set of vertex-deleted subgraphs of a graphH, that is, eachG _v ≽H −v, whereV(H)=V(G). In this case,G is a hypergraphic reconstruction ofH. We show that certain families of self-complementary graphsH can be reconstructed in this way by a hypergraphG, and thatG can be extended to a hypergraphG ^*, all of whose subsumed graphs are isomorphic toH, whereG andG ^* are self-complementary hypergraphs. In particular, the Paley graphs can be reconstructed in this way. This work was supported by an operating grant from the Natural Sciences and Engineering Research Council of Canada.     

Partial Colorings of Unimodular Hypergraphs

Combinatorial discrepancy theory asks for vertex-colorings of hypergraphs such that all hyperedges contain all colors in (as good as possible) equal quantity. Unimodular hypergraphs are good in this sense: They (and all their induced subhypergraphs as well) can be two-colored in a way that in each hyperedge the number of vertices in one color deviates from that in the other color by at most one. Note that this means that even cardinality hyperedges are perfectly balanced, whereas odd ones have a deviation of exactly one. This observation raises the question whether one can spare these deviations of one by leaving some vertices uncolored. In this work, we give a complete characterization of when this is possible.     

Implications among linkage properties in graphs

Given a graph H with vertices w₁, …, w_m, a graph G with at least m vertices is H‐linked if for every choice of vertices v₁, …, v_m in G, there is a subdivision of H in G such that v_i is the branch vertex representing w_i (for all i ). This concept generalizes the notions of k‐linked, k‐connected, and k‐ordered graphs. For graphs H₁ and H₂ with the same order that are not contained in stars, the property of being H₁‐linked implies that of being H₂‐linked if and only if H₂?H₁. The implication also holds when H₁ is obtained from H₂ by replacing an edge xy with an edge from y to a new vertex x′. Other instances of nonimplication are obtained, using a lemma that the number of vertices appearing in minimum vertex covers of a graph G is at most the vertex cover number plus the size of a maximum matching. © 2009 Wiley Periodicals, Inc. J Graph Theory 60: 327‐337, 2009     

Efficiently Recognizing the P 4-Structure of Trees and of Bipartite Graphs Without Short Cycles

A 4-uniform hypergraph represents the P ₄-structure of a graph G if its hyperedges are the vertex sets of the P ₄'s in G. By using the weighted 2-section graph of the hypergraph we propose a simple efficient algorithm to decide whether a given 4-uniform hypergraph represents the P ₄-structure of a bipartite graph without 4-cycle and 6-cycle. For trees, our algorithm is different from that given by G. Ding and has a better running time namely O(n ²) where n is the number of vertices. Revised: February 18, 1998     

RandomlyH-coverable graphs

A graph israndomly matchable if every matching of the graph is contained in a perfect matching. We generalize this notion and say that a graphG israndomly H-coverable if every set of independent subgraphs, each isomorphic toH, that does not cover the vertices ofG can be extended to a larger set of independent copies ofH. Various problems are considered for the situation whereH is a path. In particular, we characterize the graphs that are randomlyP ₃ -coverable.     

A characterization of graphs without long induced paths

In a connected graph define the k-center as the set of vertices whose distance from any other vertex is at most k. We say that a vertex set S d-dominates G if for every vertex x there is a y ∈ S whose distance from x is at most d. Call a graph P_t-free if it does not contain a path on t vertices as an induced subgraph. We prove that a connected graph is P_2k-1-free (P_2k-free) if and only if each of its connected induced subgraphs H satisfy the following property: The k-center of H (k - 1)-dominates ((k - 2)-dominates) H. Moreover, we show that the subgraph induced by the (t - 3)-center in any P_t-free connected graph is again connected and has diameter at most t - 3.     

Graphoidal bipartite graphs

A graphoidal cover of a graph G is a collection ψ of (not necessarily open) paths inG such that every path in ψ has at least two vertices, every vertex ofG is an internal vertex of at most one path in ψ and every edge of G is in exactly one path in ψ. Let Ω (ψ) denote the intersection graph of ψ. A graph G is said to be graphoidal if there exists a graphH and a graphoidal cover ψof H such that G is isomorphic to Ω(ψ). In this paper we study the properties of graphoidal graphs and obtain a forbidden subgraph characterisation of bipartite graphoidal graphs.     

On graphs with prescribed median I

The distance of a vertex u in a connected graph H is the sum of all the distances from u to the other vertices of H. The median M(H) of H is the subgraph of H induced by the vertices of minimum distance. For any graph G, let f(G) denote the minimum order of a connected graph H satisfying M(H) ? G. It is shown that if G has n vertices and minimum degree δ then f(G) ? 2n ? δ + 1. Graphs having both median and center prescribed are constructed. It is also shown that if the vertices of a K_r are removed from a graph H, then at most r components of the resulting graph contain median vertices of H.     

Super-Magic Complete k-Partite Hypergraphs

 We deal with complete k-partite hypergraphs and we show that for all k≥2 and n≠2,6 its hyperedges can be labeled by consecutive integers 1,2,…,n ^k such that the sum of labels of the hyperedges incident to (k−1) particular vertices is the same for all (k−1)-tuples of vertices from (k−1) independent sets. Received: December 8, 1997 Final version received: July 26, 1999     

On the Expected Number of Shadow Vertices of the Convex Hull of Random Points

Leta₁, . . . ,a_mbe independent random points in ⁿthat are independent and identically distributed spherically symmetrical in ⁿ. Moreover, letXbe the random polytope generated as the convex hull ofa₁, . . . ,a_mand letL_kbe an arbitraryk-dimensional subspace of ⁿwith 2 ≤k≤n− 1. LetX_kbe the orthogonal projection image ofXinL_k. We call those vertices ofXwhose projection images inL_kare vertices ofX_kshadow vertices ofXwith respect to the subspaceL_k. We derive a distribution independent sharp upper bound for the expected number of shadow vertices ofXinL_k.     

Not-all-equal 3-SAT and 2-colorings of 4-regular 4-uniform hypergraphs

In this paper, we continue our study of 2-colorings in hypergraphs (see, Henning and Yeo, 2013). A hypergraph is 2-colorable if there is a 2-coloring of the vertices with no monochromatic hyperedge. It is known (see Thomassen, 1992) that every 4-uniform 4-regular hypergraph is 2-colorable. Our main result in this paper is a strengthening of this result. For this purpose, we define a vertex in a hypergraph $H$ to be a free vertex in $H$ if we can 2-color $V\left(H\right)?\left\{v\right\}$ such that every hyperedge in $H$ contains vertices of both colors (where $v$ has no color). We prove that every 4-uniform 4-regular hypergraph has a free vertex. This proves a conjecture in Henning and Yeo (2015). Our proofs use a new result on not-all-equal 3-SAT which is also proved in this paper and is of interest in its own right.     

Contact patterns of equal nonoverlapping spheres

LetF be a set of nonoverlapping spheres in Euclideann-spaceE ⁿ . By the contact pattern ofF we mean the graph whose vertex set isF and two vertices are adjacent whenever the corresponding spheres touch each other. Every graph turns out to be a contact pattern in some dimension. This paper studies the smallest dimensionn for a graphG such thatG is a contact pattern inE ⁿ . Among others, the smallest dimensions are determined for the join of a large complete graph and an empty graph, and for complete multipartite graphs with more vertex classes than the size of its largest vertex class.     

On Switching to H‐Free Graphs

In this article, we study the problem of deciding if, for a fixed graph H, a given graph is switching equivalent to an H‐free graph. Polynomial‐time algorithms are known for H having at most three vertices or isomorphic to P₄. We show that for H isomorphic to a claw, the problem is polynomial, too. On the other hand, we give infinitely many graphs H such that the problem is NP‐complete, thus solving an open problem [Kratochvíl, Ne?et?il and Zýka, Ann Discrete Math 51 (1992)]. Further, we give a characterization of graphs switching equivalent to a K_{1, 2}‐free graph by ten forbidden‐induced subgraphs, each having five vertices. We also give the forbidden‐induced subgraphs for graphs switching equivalent to a forest of bounded vertex degrees.     

On the number of vertices and edges of the Buneman graph

In 1971, Peter Buneman proposed a way to construct a tree from a collection of pairwise compatible splits. This construction immediately generalizes to arbitrary collections of splits, and yields a connected median graph, called the Buneman graph. In this paper, we prove that the vertices and the edges of this graph can be described in a very simple way: given a collection of splitsS, the vertices of the Buneman graph correspond precisely to the subsetsS′ ofS such that the splits inS′ are pairwise incompatible and the edges correspond to pairs (S′, S) withS′ as above andS∈S′. Using this characterization, it is much more straightforward to construct the vertices of the Buneman graph than using prior constructions. We also recover as an immediate consequence of this enumeration that the Buneman graph is a tree, that is, that the number of vertices exceeds the number of edges (by one), if and only if any two distinct splits inS are compatible.