Győri, Ervin

Katona, Gyula Y.

Lemons, Nathan W.

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Hypergrpah Extensions of theErdo˝s-Gallai TheoremErvin Gyo˝ri∗Alfre´d Re´nyi Institute of MathematicsHungarian Academy of Sciencesgyori.ervin@renyi.huGyula Y. Katona∗†Department of Computer Science andInformation TheoryBudapest University of Technology andEconomicskiskat@cs.bme.huNathan W. Lemons∗Los Alamos National LabratoryCenter for Nonlinear Studiesnlemons@lanl.govAbstract: The Erdo˝s-Gallai Theorem gives the maximum number of edges in a graph withouta path of length k. We extend this result for Berge paths in r-uniform hypergraphs. We alsofind the extremal hypergraphs avoiding t-tight paths of a given length and consider thisextremal problem for other definitions of paths in hypergraphs.Keywords: extremal graph, hypergraph, Berge path1 IntroductionFor reference we state the classical result:Theorem 1 (Erdo˝s-Gallai [2]) Let G be a graph on n vertices containing no path of length k. Thene(G) ≤ 12 (k − 1)n. Equality holds iff G is the disjoint union of complete graphs on k vertices.We consider several generalizations of this theorem for hypergraphs. This is due to the fact that thereare several possible ways to define paths in hypergraphs. One such definition of paths in hypergraphs isdue to Berge.Definition 2 A Berge path of length k in a hypergraph is a collection of k hyperedges h1, . . . , hk andk + 1 vertices v1, . . . , vk+1 such that for each 1 ≤ i ≤ k we have vi, vi+1 ∈ hi.We find the extremal sizes of r-uniform hypergraphs avoiding Berge cycles of length k. Interestingly,the size of the extremal hypergraphs depend on the relationship between r and k. Specifically, wedistinguish between the cases when k ≤ r and when k > r.Theorem 3 Fix r > 2, let k > r and let H be a hypergraph containing no Berge path of length k. Thene(H) ≤ nk(kr).On the other hand, if k ≤ r, we have the following theorem.∗Research partially supported by the Hungarian National Research Fund (Grant Number 78439)†The work reported in the paper has been developed in the framework of the project “Talent care and cultivation in thescientific workshops of BME” project. This project is supported by the grant TA´MOP - 4.2.2.B-10/1–2010-00091Theorem 4 Fix 2 < k ≤ r. If H is an r-uniform hypergraph with no path of length k, then e(H) ≤n(k−1)r+1 .Remark 5 Both of the above theorems are sharp as the following two examples show. In the first case,if k > r, partition the n vertices into sets of size k. In each k set, take all possible subsets of size r tobe in the hypergraph. Such a hypergraph has exactly nk(kr)hyperedges and clearly contains no k-path asevery component is composed of exactly k vertices.In the second case, k ≤ r. Here we partition the vertices into sets of size r+ 1 and then on each r+ 1set, we select exactly k − 1 of its subsets of size r to be in the hypergraph. This hypergraph has exactlyk−1r+1n hyperedges and as each component is encompasses exactly k − 1 edges, it is clear there is no pathof length k. In this paper we will not deal with the case when k = 2, as it is trivial, but it is interestingto note that the above construction is not best possible when k = 2.We further generalize the idea of a Berge path:Definition 6 Fix r ≥ 2 and t, 1 ≤ t ≤ r − 1. A t-tight path of length k in a r-uniform hypergraph isa Berge-path on k + 1 vertices {v1, v2, . . . , vk+1} and k hyperedges {h1, h2, . . . hk} such that consecutivehyperedges intersect in at least t points.Of course a 1-tight path is the same as a Berge path. A slight variation of our t-tight paths have beenstudied before in other settings. (See for example [1].) As in the case of Berge paths, we can get quiteexact results regarding hypergraphs avoiding t-tight paths.Theorem 7 Fix r ≥ 2 and t, 1 ≤ t ≤ r − 1. Fix k large. Let H be an extremal r-uniform hypergraph onn vertices containing no t-tight path of length k. Then(1 + o(1))(ni)(kr)(kt) ≤ e(H) ≤ (nt)(kr)(kt) .The lower bound follows directly from a theorem of Ro¨dl [4].Theorem 8 The packing number m(n, k, l) = (1 + o(1))(nl)(kl). (This is the size of the largest k-uniformfamily of subsets of an n-set such that every l-set is contained in at most 1 member of the family.)Definition 9 ([5]) A tight path of length k in a r-uniform hypergraph is a collection of k+ r− 1 vertices{v1, v2, . . . , vk+r−1} and k hyperedges {h1, h2, . . . hk} such that for each 1 ≤ i ≤ k, hi = {vi, vi+1, . . . , vi+r−1}.Note that tight cycles in hypergraphs have also been studied before regarding the extremal numberfor Hamiltonian cycles. See [6] and [3].Theorem 10 Let H be an extremal r-uniform hypergraph containing no tight path of length k. Then(1 + o(1))k − 1r(nr − 1)≤ |e(H)| ≤ (k − 1)(nr − 1).Finally, note that there is an important difference between (r − 1)-tight paths and tight paths inr-uniform hypergraphs. We can investigate this difference by considering the following.Definition 11 Let k > r ≥ 2 and 1 ≤ J < k. Then a Berge path of length k in a r-uniform hypergraphH on hyperedges e1, . . . , ek satisfies interseciton conditions (J) iffor 1 ≤ l ≤ J and i > l, |ei ∩ ei−l| = max{r − l, 0}.2Of course a Berge path satisfying intersection conditions (1) is the same as an r−1 tight path. Further-more, a Berge path satisfying intersection conditions (k-1) is exactly a tight path. It is interesting that asproved above, an extremal hypergraph excluding (r− 1)-tight paths of length k contains assymptoticallyk−r+1r(nr−1)hyperedges. On the other hand, our best construction for a hypergraph excluding a tight pathof lenght k contains k−1r(nr−1)hyperedges. Supposedly, each Berge path satisfying intersection conditions(J) falls somewhere between these two. However, while there k−1 different intersection conditions, thereare only r − 1 possible block sizes in the construction of the conjectured extremal hypergraphs.The rest of the paper is organized as follows. In section 2 we prove theorem 3, the proof of 4 isomitted in this extended abstract. In section 3 we look at t-tight paths. In section 4 we consider tightpaths and (r − 1)-tight paths satisfying a certain number of intersection conditions as in Definition 11.Some of the proofs are omitted here.2 Berge PathsProof: (Proof of Theorem 3) We prove this by induction on n. Clearly, for small values of n, thetheorem trivially holds. Now, fix n such that the theorem holds for all n′ < n Then let H = (E , V ) be ar-uniform hypergraph on n vertices with e(H) > nk(kr). We can assume that the following holds for theminimal degree, δ, in H.δ = e(H) rn>1r(k − 1r − 1)(1)Otherwise, if there is a vertex x in H with degree no more than 1r(k−1r−1), then we may delete this vertex(and all the edges incident with it) from H. The result will be a hypergraph on n− 1 vertices with morethan n−1k(kr)edges. Thus by the induction hypothesis, this hypergraph will have a path of length k.Let P be the longest path in H. Let v1, v2, . . . , vl+1 be the vertices of P , and h1, h2, . . . , hl thehyperedges such that for each i, vi, vi+1 ∈ hi. Suppose that l < k.We claim that if there is a cycle of length l + 1 on the vertices v1, v2, . . . , vl+1, then these verticesconstitute a component of the hypergraph H. To see this, suppose that C is such a cycle. Then if an edgeh in the cycle C does not lie completely within the vertices v1, v2, . . . , vl+1, then deleting h from C wehave an l- path which can be extended (by the edge h) to a path of length at least l+ 1. Thus every edgeh in the cycle C must be contained within the vertices v1, v2, . . . , vl+1. In fact, something stronger is true.For each vertex in the cycle, vi, the neighborhood of vi lies within v1, v2, . . . , vl+1. (The neighborhood ofa vertex is the set of vertices in H which are connected to vi by an edge.) Indeed, suppose that for somei, the vertex vi has a neighbor y outside of {v1, v2, . . . , vl+1}. Then the edge containing both vi and y isnot an edge of C (by the above argument.) Thus, removing an appropriate edge of C so that it is a pathof length l with vi as an endpoint, we can extend this to a path of length l + 1 with y as an endpoint, acontradiction. This proves our claim.In the remainder of the proof we show that there must be an l+1 cycle on the vertices v1, v2, . . . , vl+1.As this constitutes a connected component of H with at most (l+1r ) ≤ (kr) hyperedges, we may delete thiscomponent from H and by the induction hypothesis find a path of length k on what remains.It remains to show that the vertices v1, v2, . . . , vl+1 form a cycle of length l + 1. Let H′ be thehypergraph obtained by deleting the edges of P from H. Specifically, let H′ = H\{h1, h2, . . . , hl}. Notethat by the choice of P , the neighborhoods of v1 and vl+1 in H′ must fall within {v1, v2, . . . , vl+1}Clearly, if there is an edge of H′ containing both v1 and vl+1, then the edges of P form a cycle oflength l + 1. On the other hand, if there exist edges g1, g2 ∈ H′ such that for some i, 1 < i < l+,v1, vi+1 ∈ g1 and vl + 1, vi ∈ g2 then there is an l + 1 cycle, on the verticesv1, vi+1, vi+2, vi+3, . . . , vl+1, vi, vi−1, vi−2, . . . , v1.Thus by the pigeon hole principle, if in H′ the degrees of both v1 and vl+1 are greater then( k−22r−1), thenthere is a l + 1 cycle on v1, v2, . . . , vl+1 in H.3We will show that this is in fact true; that the minimum degree in H′ is greater than ( k−22r−1). Of course,it is sufficient to show that the minimum degree in H is greater than ( k−22r−1)+k−1. Indeed from Equation(1) it is enough to show that the following holds.(k − 1r − 1)≥( k−22r − 1)+ k − 1 (2)We leave it as an exercise for the reader to verify that Inequality (2) holds for all k > r ≥ 3.This completes the proof of the theorem. 23 t-Tight PathsProof: (Proof of Theorem 7) First consider the lower bound. By Theorem 8, there is a family B ofk-sets of an initial n set such that no t set is contained in more than 1 element of the family and suchthat|B| ≥ (1 + o(1))(nt)(kt) .We claim that the r-uniform hypergraph H obtained by replacing each member of B with all its (kr)r-sets contains no t-tight path. Any such path would have vertices in at least 2 different members of B.Specifically, such a path would contain vertices u and v with u ∈ B1\B2 and v ∈ B2\B1 where B1 andB2 are two distinct members of B. But |B1 ∩ B2| < t (the same holds for all distinct pairs of membersof B.) Thus there can be no t-tight path in H from u to v.We now look at the upper bound. If t = 1 then we are done by Theorem 3. Suppose then that t ≥ 2.Let H be a hypergraph on n vertices with more than (nt)(kr)/(kt) hyperedges. Then it is easy to see thatthere exists a vertex x1 ∈ V with (nt)(kr)(kt) = (n−1t−1)(k−1r−1)(k−1t−1)Let H1 = {h\{x1} : h ∈ H} be the link of x1. Then continuing we can clearly find vertices x2, . . . , xt−1such that for 1 < i < t, Hi is the link of xi in Hi−1 and such that for 1 < i < t, the degree of xi in Hi−1is greater than(n−it−i)(k−ir−i)/(k−it−i). But then Ht−1 is simply a (r − t+ 1)-graph on n− t+ 1 vertices withmore than n−t+1k−t+1(k−t+1r−t+1)edges. But then applying Theorem 3 we can find a path of length k − t+ 1 inHt−1. If the minimal degree in Ht−1 is large enough, we can then extend this path using the verticesx1, . . . , xt−1 to a t-tight path of length k in H. 24 Even Tighter PathsIn this section we consider the relationship between tight and t-tight paths as mentioned in Definition11. We prove Theorem 10 and a related theorem concerning (r − 1)-tight paths satisfying intersectionconditions J for fixed 1 ≤ J ≤ k − 1. First we will need a simple averaging argument.Lemma 12 Let H be an r-uniform hypergraph on n vertices with c( nr−1) edges. Then there exists annonempty sub-hypergraph, H′ , of H such that∀S ∈(V(H)r − 1), dH′ (S) ≤ c⇒ dH′ (S) = 0 (3)where dH′ (S) refers to the number of hyperedges of H′containing the set S.4Proof: (Proof of Theorem 10) The lower bound of course follows from our usual construction usingTheorem 8. The upper bound follows just as easily from Lemma 12. If an r-uniform hypergraph H on nvertices satisfies 3 then it is quite clear that we can find a tight path of length k in H: in fact every edgeof H will be contained in such a path. 2It is thus perhaps unsurprising that the upper bound (and trivially the lower bound) of Theorem 7 for(r−1)-tight paths also holds for paths satisfying intersection conditions (2) if k is big enough compared tor. On the other hand it is easy to see that our construction for maximal hypergraphs containing no tightpath does indeed contain Berge paths satisfying intersection conditions (k − 2). We give the followingtheorem which is clearly best possible up to a factor of r.Theorem 13 Let H be an r-uniform hypergraph containing no Berge path of length k which satisfiesintersection conditions (J). (Where r, k, J are as in Def. 11.) If k− J > r− 1, set a := r− 1. Otherwiseset a := k − J . Then e(H) < (k − a)( nr−1).Note that for J = 1 (i.e. for (r − 1)-tight paths), the theorem returns a weaker result than Theorem7.References[1] Dorbec, P., Gravier, S and Sa´rko¨zy, G. , Monochromatic Hamiltonian t-tight Berge-cyclesin hypergraphs, Journal of Graph Theory 59, n. 1 (2008) 34-44.[2] Erdo˝s, P., and Gallai, T. , On maximal paths and circuits of graphs, Acta Math. Acad. Sci.Hungar. 10, (1959) 337-356.[3] Frankl, P. and Katona, G. Y. , Extremal k-edge Hamiltonian hypergraphs, Discrete Math.308 n. 8 (2008) 1415-1424.[4] Ro¨dl, V. , On a packing and covering problem, European J. of Combinatorics 6, (1985) 69-78.[5] G. Y. Katona and H. A. Kierstead, Hamiltonian chains in hypergraphs, J. Graph Theory,30(3) (1999) 205–212.[6] Ro¨dl, V., Rucin´ski, A. and Szemere´di, E. , An approximate Dirac-type theorem for k-uniformhypergaphs, Combinatorica 28, n. 2 (2008) 229-260.5

Hypergrpah Extensions of the Erdős-Gallai Theorem

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