Powers of Hamilton cycles of high discrepancy are unavoidable
aa r X i v : . [ m a t h . C O ] F e b Powers of Hamilton cycles of high discrepancy are unavoidable
Domagoj BradaËc â March 1, 2021
Abstract
The PÂŽosa-Seymour conjecture asserts that every graph on ð vertices with minimum degree at least ( â /( ð + )) ð contains the ð ð¡â power of a Hamilton cycle. KomlÂŽos, SÂŽarkšozy and SzemerÂŽedi famouslyproved the conjecture for large ð. The notion of discrepancy appears in many areas of mathematics,including graph theory. In this setting, a graph ðº is given along with a 2-coloring of its edges. Oneis then asked to ï¬nd in ðº a copy of a given subgraph with a large discrepancy, i.e., with many moreedges in one of the colors. For ð ⥠, we determine the minimum degree threshold needed to ï¬ndthe ð ð¡â power of a Hamilton cycle of large discrepancy, answering a question posed by Balogh, Csaba,PluhÂŽar and Treglown. Notably, for ð ⥠, this threshold approximately matches the minimum degreerequirement of the PÂŽosa-Seymour conjecture. Classical discrepancy theory studies problems of the following kind: given a family of subsets of a universalset U , is it possible to partition the elements of U into two parts such that each set in the family hasroughly the same number of elements from each part? One of the ï¬rst signiï¬cant results in this area is acriterion for a sequence to be uniformly distributed in the unit interval proved by Hermann Weyl. Sincethen, discrepancy theory has had wide applicability in many ï¬elds such as ergodic theory, number theory,statistics, geometry, computer science, etc. For a comprehensive overview of the ï¬eld, see the books byBeck and Chen [3] and by Chazelle [5].This paper studies a problem in the discrepancy theory of graphs. To discuss the topic, we start with adeï¬nition. Deï¬nition 1.1.
Let ðº = ( ð , ðž ) be a graph and ð : ðž â {â , } a labelling of its edges. Given a subgraph ð» of ðº, we deï¬ne its discrepancy ð ( ð» ) as ð ( ð» ) = à ð â ðž ( ð» ) ð ( ð ) . Furthermore, we refer to the value | ð ( ð» ) | as the absolute discrepancy of ð» .
One of the central questions in graph discrepancy theory is the following. Suppose we are given a graph ðº and a spanning subgraph ð» .
Does ðº, for every edge labelling ð : ðž ( ðº ) â {â , } , contain an isomorphiccopy of ð» of high absolute discrepancy with respect to ð ? ErdËos, Fšuredi, Loebl and SÂŽos [6] proved the ï¬rstresult of this kind. They show that, given a tree on ð vertices ð ð with maximum degree Î and a {â , } -coloring of the edges of the complete graph ðŸ ð , one can ï¬nd a copy of ð ð with absolute discrepancy at least ð ( ð â â Î ) , for some absolute constant ð > . â Department of Computer Science, ETH Zšurich, 8092 Zšurich, Switzerland. Email: [email protected] commonly studied topic in extremal combinatorics are
Dirac-type problems where one is given a graph ðº on ð vertices with minimum degree at least ðŒð and wants to prove that ðº contains a copy of a speciï¬cspanning subgraph ð» . In the discrepancy setting it is natural to ask whether we can also ï¬nd a copy of ð» with large absolute discrepancy. Balogh, Csaba, Jing and PluhÂŽar studied this problem for spanning trees,paths and Hamilton cycles. Among other results, they determine the minimum degree threshold neededto force a Hamilton cycle of high discrepancy. Theorem 1.2 (Balogh, Csaba, Jing and PluhÂŽar [1]) . Let < ð < / and ð â N be suï¬ciently large. If ðº isan ð -vertex graph with ð¿ ( ðº ) ⥠( / + ð ) ð and ð : ðž ( ðº ) â {â , } , then ðº contains a Hamilton cycle with absolute discrepancy at least ðð / withrespect to ð . Moreover, if divides ð, there is an ð -vertex graph with ð¿ ( ðº ) = ð / and an edge labelling ð : ðž ( ðº ) â {â , } such that any Hamilton cycle in ðº has discrepancy with respect to ð . Very recently, Freschi, Hyde, Lada and Treglown [7], and independently, Gishboliner, Krivelevich andMichaeli [8] generalized this result to edge-colorings with more than two colors.A fundamental result in extremal graph theory is the Hajnal-SzemerÂŽedi theorem. It states that if ð divides ð and ðº is a graph on ð vertices with ð¿ ( ðº ) ⥠( â / ð ) ð, then ðº contains a perfect ðŸ ð -tiling, i.e. its vertexset can be partitioned into disjoint cliques of size ð . Balogh, Csaba, PluhÂŽar and Treglown [2] proved adiscrepancy version of this theorem.
Theorem 1.3 (Balogh, Csaba, PluhÂŽar and Treglown [2]) . Suppose ð ⥠is an integer and let ð > . Thenthere exists ð â N and ðŸ > such that the following holds. Let ðº be a graph on ð ⥠ð vertices where ð divides ð and where ð¿ ( ðº ) ⥠(cid:18) â ð + + ð (cid:19) ð. Given any function ð : ðž ( ðº ) â {â , } there exists a perfect ðŸ ð -tiling T in ðº so that (cid:12)(cid:12)(cid:12) à ð â ðž (T) ð ( ð ) (cid:12)(cid:12)(cid:12) ⥠ðŸð. The ð ð¡â power of a graph ðº is the graph on the same vertex set in which two vertices are joined by an edge ifand only if their distance in ðº is at most ð . The PÂŽosa-Seymour conjecture asserts that any graph on ð verticeswith minimum degree at least ( â /( ð + )) ð contains the ð ð¡â power of a Hamilton cycle. KomlÂŽos, SÂŽarkšozyand SzemerÂŽedi [12] proved the conjecture for large ð. In [2] the authors posed the question of determiningthe minimum degree needed to force the ð ð¡â power of a Hamilton cycle with absolute discrepancy linearin ð. Because the ð ð¡â power of a Hamilton cycle contains a (almost) perfect ( ð + ) -tiling, they suggestedthe minimum degree required should be ( â /( ð + ) + ð ) ð, based on their result for ðŸ ð -tilings. We provethis value is correct for ð = . However, we show that for ð ⥠, a minimum degree of ( â /( ð + ) + ð ) ð, for arbitrarily small ð > , is suï¬cient, approximately matching the minimum degree required for ï¬ndingany ð ð¡â power of a Hamilton cycle. As far as the author knows, this is the ï¬rst Dirac-type discrepancyresult in which the threshold for ï¬nding a spanning subgraph of large discrepancy is the same, up to anarbitrarily small linear term, as the minimum degree required for ï¬nding any copy of the subgraph. Theorem 1.4.
For any integer ð ⥠and ð > , there exist ð â N and ðŸ > such that the following holds.Suppose a graph ðº on ð ⥠ð vertices with minimum degree ð¿ ( ðº ) ⥠( â /( ð + ) + ð ) ð and an edge coloring ð : ðž ( ðº ) â {â , } are given. Then in ðº there exists the ð ð¡â power of a Hamilton cycle ð» ð satisfying (cid:12)(cid:12)(cid:12) à ð â ðž ( ð» ð ) ð ( ð ) (cid:12)(cid:12)(cid:12) ⥠ðŸð. ð ð¡â power of a Hamilton cycle of large discrepancyis the same for ð â { , , } and equals ( + ð ) ð. The cases ð = , ð = . Theorem 1.5.
For any ð > , there exist ð â N and ðŸ > such that the following holds. Suppose a graph ðº on ð ⥠ð vertices with minimum degree ð¿ ( ðº ) ⥠( / + ð ) ð and an edge coloring ð : ðž ( ðº ) â {â , } aregiven. Then in ðº there exists the square of a Hamilton cycle ð» satisfying (cid:12)(cid:12)(cid:12) à ð â ðž ( ð» ) ð ( ð ) (cid:12)(cid:12)(cid:12) ⥠ðŸð. These results are tight in the following sense. If we weaken the minimum degree requirement by replacingthe term ðð with a sublinear term, then there are examples in which any ð ð¡â power of a Hamilton cyclehas absolute discrepancy ð ( ð ) . The paper is organised as follows. In Section 2 we introduce some notation and deï¬nitions and stateprevious results used in our proofs. Then we present lower bounds in Section 3. We give a short outline ofthe proofs in Section 4. The proofs are then divided into two sections. In Section 5 we adapt the proof byKomlÂŽos, SÂŽarkšozy and SzemerÂŽedi of an approximate version of the PÂŽosa-Seymour conjecture to our setting,while the rest of the argument is presented in Section 6.
Most of the graph theory notation we use is standard in the literature and can be found in [4]. Let ðº be agraph. We use ð ðº ( ð£ ) to denote the neighbourhood of ð£ in ðº .
Given an edge labelling ð : ðž ( ðº ) â {â , } , we use ðº + to denote the graph containing all edges labelled 1 and ðº â to denote the graph containing alledges labelled â . We write ð + ðº ( ð£ ) for the set of ð¢ in ð ðº ( ð£ ) such that ð ( ð£, ð¢ ) = ð â ðº ( ð£ ) for the set of ð¢ in ð ðº ( ð£ ) such that ð ( ð£, ð¢ ) = â . For a vertex ð£ and a subset of vertices ð , we deï¬ne ð ( ð£, ð ) = ð ( ð£ ) â© ð and deg ( ð£, ð ) = | ð ( ð£, ð ) | . We write . ⪠for the union of disjoint sets. We use the terms edge labelling andedge coloring interchangeably. We sometimes omit the underlying graph when it is clear from the context.Throughout the paper we allow cycles to have repeated vertices, unless explicitly stated they are simple.We deï¬ne the multiplicity mul ð¶ ( ð£ ) of a vertex ð£ in a cycle ð¶ as the number of occurences of ð£ in ð¶ whenviewed as a closed walk. Given a cycle ð¶ we deï¬ne its ð ð¡â power as the multigraph obtained by connectingvertices at every pair of indices at most ð apart in the cyclic order deï¬ning ð¶. More formally, for a cycle ð¶ = ( ð£ , ð£ , . . . , ð£ ð ) , we deï¬ne its ð ð¡â power, denoted by ð¶ ð = ( ð£ , ð£ , . . . , ð£ ð ) ð , as the multigraph with thefollowing edge multiplicities:mul ð¶ ð ( ð¥ðŠ ) = (cid:12)(cid:12)(cid:8) ( ð, ð ) | ð â [ ð ] , ð â [ ð ] , { ð£ ð , ð£ ð + ð } = { ð¥, ðŠ } (cid:9)(cid:12)(cid:12) , where we denote ð£ ð + ð = ð£ ð for 1 †ð †ð . The ð ð¡â power of a simple ( ð + ) -cycle will sometimes be referredto as an ( ð + ) -clique. Importantly, though, it has two copies of each edge. Given an edge labelling ð : ðž ( ðº ) â {â , } , we deï¬ne the discrepancy of ð¶ ð in the natural way: ð ( ð¶ ð ) = à ð â ðž ( ð¶ ð ) mul ð¶ ð ( ð ) ð ( ð ) . Similarly as in [2], we deï¬ne a ð¶ ð -template. Note that in the following deï¬nition we only allow shortcycles. Deï¬nition 2.1.
Let ð¹ be a graph. A ð¶ ð -template of ð¹ is a collection F = { ð¶ , ð¶ , . . . , ð¶ ð } of not necessarilydistinct cycles whose ð ð¡â powers are subgraphs of ð¹ .
In a ð¶ ð -template each vertex appears the same numberof times, that is, à ð ð = mul ð¶ ð ( ð£ ) is the same for all ð£ â ð ( ð¹ ) . Moreover, we require that each cycle ð¶ ð haslength between ð + ð . The discrepancy of a ð¶ ð -template is given as ð (F ) = à ð ð = ð ( ð¶ ðð ) .
3e notion of a ð¶ ð -tiling is obtained by adding the natural restriction that each vertex appears exactlyonce. Deï¬nition 2.2.
Let ð¹ be a graph. A ð¶ ð -tiling T of ð¹ is a collection of simple cycles T = { ð¶ , ð¶ , . . . , ð¶ ð } whose ð ð¡â powers are subgraphs of ð¹ and each vertex appears precisely once in these cycles. The lengthof each cycle is between ð + ð . The discrepancy of a ð¶ ð -tiling is given as ð (T ) = à ð ð = ð ( ð¶ ðð ) . In the above two deï¬nitions, we slightly abuse notation in the following sense. We ignore edge multiplicitiesfor the notion of graph containment (as the ambient graph is always simple). In other words, we onlyrequire the ambient graph to have one copy of each edge that has positive multiplicity in a given ð ð¡â power of a cycle. A ðŸ ð + -tiling is a ð¶ ð -tiling in which all cycles have length ð + , that is, all tiles are ( ð + ) -cliques.We give names to special types of ð -cliques with respect to an edge labelling ð . Deï¬nition 2.3.
We write ðŸ + ð for the ð -clique with all edges labelled 1 and ðŸ â ð for the ð -clique with alledges labelled â . The ( ðŸ ð , +) -star is the clique whose edges labelled 1 induce a copy of ðŸ ,ð â . The root ofthis ðŸ ,ð â is called the head of the ( ðŸ ð , +) -star. We deï¬ne the ( ðŸ ð , â) -star and its head analogously.We write ðŒ ⪠ðœ ⪠ðŸ, if the constants can be chosen from right to left such that all calculations in our proofare valid. More precisely, ðŒ ⪠ðœ means there is a positive increasing function ð ( ðœ ) such that for ðŒ = ð ( ðœ ) , all calculations in the proof are valid. This notion naturally extends to hierarchies of larger length as well.We omit ï¬oors and ceilings whenever they do not aï¬ect the argument.In our proofs we use the famous Hajnal-SzemerÂŽedi theorem in the following form. Theorem 2.4 (Hajnal and SzemerÂŽedi [9]) . Every graph ðº whose order ð is divisible by ð and has minimumdegree at least ( â / ð ) ð contains a perfect ðŸ ð -tiling. In the proof of the main result, we use a multicolored variant of SzemerÂŽediâs regularity lemma [14]. Beforestating the result, we deï¬ne the relevant notions. The density of a bipartite graph ðº with vertex classes ðŽ and ðµ is deï¬ned as ð ( ðŽ, ðµ ) = ð ( ðŽ, ðµ )| ðŽ || ðµ | . Given ð, ð > , the graph ðº is said to be ( ð, ð ) -regular if ð ( ðŽ, ðµ ) ⥠ð and for any ð â ðŽ, ð â ðµ such that | ð | > ð | ðŽ | and | ð | > ð | ðµ | , we have | ð ( ðŽ, ðµ ) â ð ( ð, ð ) | < ð. The graph ðº is ( ð, ð¿ ) -super-regular if for every ð â ðŽ with | ð | > ð | ðŽ | and ð â ðµ with | ð | > ð | ðµ | , we have ð ( ð, ð ) > ð¿, and furthermore, deg ( ð ) > ð¿ | ðµ | for all ð â ðŽ and deg ( ð ) > ð¿ | ðŽ | for all ð â ðµ. Suppose we are given a graph ðº with an edge labelling ð : ðž ( ðº ) â {â , } . Given disjoint
ð, ð â ð ( ðº ) we write ( ð, ð ) + ðº or ðº + [ ð, ð ] for the bipartite graphwith vertex classes ð, ð containing edges between ð and ð labelled 1 . Analogously, we deï¬ne ( ð, ð ) â ðº and ðº â [ ð, ð ] with respect to edges labelled â . We use a variant of the regularity lemma which is easily deduced from the multicolored version in [13].
Lemma 2.5.
For every ð > and ð, â â N there exists ð¿ = ð¿ ( ð, â ) such that the following holds. Let ð â [ , ) and let ðº be a graph on ð ⥠ð¿ vertices with an edge coloring ð : ðž ( ðº ) â {â , } . Then, there existsa partition ( ð ð ) âð = , for some â â [ â , ð¿ ] divisible by ð, of ð ( ðº ) and a spanning subgraph ðº â² of ðº with thefollowing properties:(i) | ð | †ðð and | ð | = | ð | = · · · = | ð â | , ii) deg ðº â² ( ð£ ) ⥠deg ðº ( ð£ ) â ( ð + ð ) ð for every ð£ â ð ( ðº ) ; (iii) ð ( ðº â² [ ð ð ]) = for all †ð †â ; (iv) for all †ð < ð †â and ð â {+ , â} , either ( ð ð , ð ð ) ððº â² is an ( ð, ð ) -regular pair or ðº â² ð [ ð ð , ð ð ] is empty; We call ð , . . . , ð â clusters and ð the exceptional set . We refer to ðº â² as the pure graph . We deï¬ne the reducedgraph ð of ðº with parameters ð, ð, ð, â to be the graph whose vertices are ð , . . . , ð â and where ( ð ð , ð ð ) isan edge if at least one of ( ð ð , ð ð ) + ðº â² and ( ð ð , ð ð ) â ðº â² is ( ð, ð ) -regular. On the reduced graph ð , we deï¬ne theedge coloring ð ð : ðž ( ð ) â {â , } as follows: ð ð ( ð ð , ð ð ) = ( , if ( ð ð , ð ð ) + ðº â² is ( ð, ð ) -regular â , otherwise. (1)Note that if both ( ð ð , ð ð ) + ðº â² and ( ð ð , ð ð ) â ðº â² are ( ð, ð ) -regular, ð ð only records the former property.We use a well-known fact about the reduced graph: Fact 2.6.
Let ð > be a given constant and ðº a graph on ð vertices such that ð¿ ( ðº ) ⥠ðð. Let ð be the reducedgraph obtained after applying Lemma 2.5 with parameters ð, ð and â . Then ð¿ ( ð ) ⥠( ð â ð â ð ) | ð | . The so-called Slicing Lemma states that large subsets of a regular pair are also regular with a slightly worsedegree of regularity.
Lemma 2.7 (Slicing Lemma [13]) . Let ( ðŽ, ðµ ) be an ð -regular pair with density ð, and, for some ðŒ > ð, let ðŽ â² â ðŽ, | ðŽ | Ⲡ⥠ðŒ | ðŽ | , ðµ â² â ðµ, | ðµ â² | ⥠ðŒ | ðµ | . Then ( ðŽ â² , ðµ â² ) is an ð â² -regular pair with ð â² = ððð¥ { ð / ðŒ, ð } and forits density ð â² we have | ð â² â ð | < ð. We also need the incredibly useful result of KomlÂŽos, SÂŽarkšozy and SzemerÂŽedi, known as the
Blow-up lemma,which states that ( ð, ð ) -superregular pairs behave like complete bipartite graphs in terms of containgsubgraphs of bounded degree. Lemma 2.8 (Blow-up lemma [10]) . Given a graph ð of order ð and positive parameters ð¿, Î , there exists apositive ð = ð ( ð¿, Î , ð ) such that the following holds. Let ð , ð , . . . , ð ð be arbitrary positive integers and let usreplace the vertices ð£ , ð£ , . . . , ð£ ð of ð with pairwise disjoint sets ð , ð , . . . , ð ð of sizes ð , ð , . . . , ð ð (blowingup). We construct two graphs on the same vertex set ð = ⪠ð ð . The ï¬rst graph ð¹ is obtained by replacing eachedge { ð£ ð , ð£ ð } of ð with the complete bipartite graph between the corresponding vertex-sets ð ð and ð ð . A sparsergraph ðº is constructed by replacing each edge { ð£ ð , ð£ ð } arbitrarily with an ( ð, ð¿ ) -super-regular pair between ð ð and ð ð . If a graph ð» with Î ( ð» ) †Πis embeddable into ð¹ then it is already embeddable into ðº .
The following remark also appears in [10].
Remark.
When using the Blow-up Lemma, we usually need the following strengthened version: Given ð > , there are positive numbers ð = ð ( ð¿, Î , ð, ð ) and ðŒ = ðŒ ( ð¿, Î , ð, ð ) such that the Blow-up Lemma in the equalsize case (all | ð ð | are the same) remains true if for every ð there are certain vertices ð¥ to be embedded into ð ð whose images are a priori restricted to certain sets ð¶ ð¥ â ð ð provided that(i) each ð¶ ð¥ within a ð ð is of size at least ð | ð ð | , (ii) the number of such restrictions within a ð ð is not more than ðŒ | ð ð | . We present simple lower bound constructions showing our results are best possible in a certain sense. For ð = ð ( ð ) = ð ( ) , the condition ð¿ ( ðº ) ⥠( â ð + + ð ) ð when ð ⥠ð¿ ( ðº ) ⥠( / + ð ) ð when ð =
2, is not5nough to guarantee an ð ð¡â power of a Hamilton cycle with absolute discrepancy linear in ð. Moreover,for ð = , there exists a graph in which every ð ð¡â power of a Hamilton cycle has discrepancy 0 . First consider ð ⥠. We construct a graph ðº as follows. Let ð¡ be even and ð , . . . , ð ð + disjoint clusters ofsize ð¡ . Additionally, let ð be a cluster of size ð. We construct a graph on the vertex set ð = . à ð + ð = ð ð . Weput an edge between any two vertices from diï¬erent clusters and we put all edges connecting two verticesin ð . Let ð = | ð | = ( ð + ) ð¡ + ð and note that ð¿ ( ðº ) = ðð¡ + ð = (cid:16) â ð + + ð ( ð + ) (( ð + ) ð¡ + ð ) (cid:17) ð. Next we describe the coloring ð of the edges. We color the edges incident to vertices in ð arbitrarily. Foreach ð ð , ð ⥠positive and the other half as negative . For a vertex ð£ â ð ð and any vertex ð¢ â ð ð where 1 †ð < ð we set ð ( ð£, ð¢ ) = ð£ is positive and ð ( ð£, ð¢ ) = â ð£ is negative.Let ð» ð be an ð ð¡â power of a Hamilton cycle in ðº viewed as a 2 ð -regular subgraph of ðº .
Call a vertex ð£ â ð \ ð a bad vertex if at least one of its neighbours in ð» ð is from the cluster ð , otherwise call it good . If a vertex ð£ â ð ð is good then in ð» ð it has precisely two neighbours from each of the clusters ð ð , †ð †ð + , ð â ð. Note that for ð ⥠ð ð can be adjacent to a vertex ð£ â ð , so there are at most 2 ð bad vertices in ð ð . Now consider only positive good vertices and their edges towards vertices from clusterswith a smaller index. Thus, the number of edges labelled 1 in ð» ð is at least ð + à ð = ( ð â ) ( ð¡ / â ð ) = ð ( ð + ) ( ð¡ / â ð ) . Hence, we have ð ( ð» ð ) ⥠â ðð + ð ( ð + ) ( ð¡ / â ð ) ⥠â ð ( ð + ) ð. Completely analogously, ð ( ð» ð ) †ð ( ð + ) ð. Therefore, if ð = , we have ð ( ð» ð ) = ð = ð ( ð ) , weget | ð ( ð» ð ) | = ð ( ð ) . For ð = , the following construction was given in [1], where the case ð = ðº be the4-partite TurÂŽan graph on ð = ð vertices, so ð¿ ( ðº ) = ð. Colour all edges incident to one of the parts with â . Any square of a Hamilton cycle contains 4 ð edges labelled â
1, exactly 4 for eachvertex in the special class. As it has a total of 8 ð edges, its discrepancy is 0 . Similarly as above, we can add ð = ð ( ð ) vertices connected to every other vertex and still any square of a Hamilton cycle has absolutediscrepancy ð ( ð ) . Our proof follows a very similar structure to that of Balogh, Csaba, PluhÂŽar and Treglown [2] for ðŸ ð -tilings.We start by applying the regularity lemma on ðº to obtain the reduced graph ð and the corresponding edgelabelling ð ð . Before proving the conjecture for large ð, KomlÂŽos, SÂŽarkšozy and SzemerÂŽedi [11], proved an approximateversion, namely they proved it for ð -vertex graphs with minimum degree at least ( â /( ð + ) + ð ) ð. Wemake slight modiï¬cations to their proof to establish two important claims.We prove that a ðŸ ð + -tiling of ð with linear discrepancy with respect to ð ð can be used to construct an ð ð¡â power of a Hamilton cycle in ðº with linear discrepancy with respect to ð . Combined with Theorem 1.3,this is enough to deduce the case ð = ð : suppose that ð¹ is a small subgraph of ð and there are two ð¶ ð -templates of ð¹ covering the vertices the same number of times, but having diï¬erent discrepancies with6espect to ð ð . Then in ðº there exists the ð ð¡â power of a Hamilton cycle with linear discrepancy with respectto ð . From this point on, we only âworkâ on the reduced graph ð . To use the last claim, we need a subgraph ð¹ onwhich we can ï¬nd two diï¬erent ð¶ ð -templates, so the simplest subgraph we can study is an ( ð + ) -clique.We prove that every ( ð + ) -clique in ð is either a copy of ðŸ + ð + , ðŸ â ð + , ( ðŸ ð + , +) -star or ( ðŸ ð + , â) -star. As ð has large minimum degree, every clique of size ð †ð + ð + . Thisshows that every clique of size ð †ð + ðŸ + ð , ðŸ â ð , ( ðŸ ð , +) -star or ( ðŸ ð , â) -star.We assume ðº has no ð ð¡â power of a Hamilton cycle with large absolute discrepancy. By the Hajnal-SzemerÂŽedi theorem, we can ï¬nd a ðŸ ð + -tiling T of ð . The previous arguments show that only four typesof cliques appear in T and T has a small discrepancy with respect to ð ð . This tells us that the numbers ofeach of the four types of cliques in T are balanced in some way.We consider two cliques in T of diï¬erent types. For several relevant cases when there are many edgesbetween the two cliques, we construct two ð¶ ð -templates of diï¬erent discrepancies, which contradicts ourassumption by the claim about ð¶ ð -templates. Finally, this restricts the number of edges between diï¬erentcliques which leads to a contradiction with the minimum degree assumption on ð . ð¶ ð -tilings of ð In [11] KomlÂŽos, SÂŽarkšozy and SzemerÂŽedi proved an approximate version of the PÂŽosa-Seymour conjecture.More precisely, they show that for any ð > ðº on ð vertices, for suï¬ciently large ð , with minimumdegree at least ( ð /( ð + )+ ð ) ð contains the ð ð¡â power of a Hamilton cycle. Their argument starts by applyingthe regularity lemma to get an ( ð, ð / ) -super-regular partition ( ð ð ) ð¡ð = of ð ( ðº ) and the Hajnal-SzemerÂŽeditheorem to obtain a ðŸ ð + -tiling of its ( ð, ð / ) -reduced graph ð . Let ðŸ , . . . , ðŸ ð be the ( ð + ) -cliques in thistiling. Then they proceed to ï¬nd short paths ð , . . . , ð ð in ðº , each ð ð âconnectingâ subsequent cliques ðŸ ð , ðŸ ð + in the tiling, and âattachâ the exceptional verticesâthe ones in ð together with some other vertices notrespecting a certain degree conditionâto these paths. Finally, for each clique ðŸ ð , the Blow-up Lemma [10]is applied to ï¬nd the ð ð¡â power of a Hamilton path on the set of unused vertices in ðº corresponding to thisclique, which together with paths ð , . . . , ð ð closes the ð ð¡â power of a Hamilton cycle.We need a slightly more general result for our application. Instead of a ðŸ ð + -tiling in ð , we assume a ð¶ ð -tiling T of ð is given in which most of the cycles ð¶ ð â T have length ð +
1. Then we proceed similarlyas outlined above, where for each ð¶ ð â T we choose an ( ð + ) -clique in ð¶ ðð to represent ðŸ ð for which theconnecting path ð ð is constructed. In order to argue about the discrepancy of the found ð ð¡â power of aHamilton cycle ð» ð in ðº , we explicitly state a property of the construction in [11], which is that most ofthe edges in ð» ð come from the given tiling and that these edges are used in a balanced way. Intuitively, ifthe ð¶ ð -tiling of ð is given together with a function ð ð as in (1), then the discrepancy of the found ð» ð in ðº is at most ðŒð away from ð · ð ð (T ) , for an arbitrarily small ðŒ >
0, where we remind the reader that ð isthe size of each cluster. Additionally, we show that, given two similar ð¶ ð -tilings, we can ï¬nd similar ð ð¡â powers of a Hamilton cycle. We make these notions precise in the following statement. Proposition 5.1.
For any integer ð ⥠and any ðŒ, ð > there exist â â N and ð > such that the followingholds. Suppose ðº is a graph on ð vertices with ð¿ ( ðº ) ⥠( ð /( ð + ) + ð ) ð and let ð : ðž ( ðº ) â {â , } be itsedge labelling. Let ( ð ð ) âð = , where â ⥠â and ( ð + ) | â, be an ( ð, ð / ) -super-regular partition of ð ( ðº ) and ð its ( ð, ð / ) -reduced graph with ð ð : ðž ( ð ) â {â , } as deï¬ned in (1) . Suppose we are given two ð¶ ð -tilings T = K . ⪠C and T = K . ⪠C of ð such that ⢠K consists only of ( ð + ) -cycles, and |C ð | †ð , for ð â { , } .Then there exist ð ð¡â powers of Hamilton cycles ð» ð , ð» ð â ðº such that(i) | ð ( ð» ðð ) â ðð ð (T ð ) | †ðŒð , for ð â { , } , and(ii) | ð ( ð» ð ) â ð ( ð» ð ) | ⥠ð | ð ð (C ) â ð ð (C ) | â ð¶ð ,where ð¶ > is an absolute constant.Proof. The proof is almost a one-to-one copy of the argument by KomlÂŽos, SÂŽarkšozy and SzemerÂŽedi in [11].We thus only give a rough sketch without spelling out all the details but speciï¬cally list the necessarychanges to adapt it to our setting. It follows the rough outline given in the paragraphs above the statement.We may assume 1 / ð ⪠ðŒ ⪠/ â ⪠ð ⪠ð ⪠/ ð . Recall that | ð | †ðð and | ð ð | = ð for 1 †ð †â. Let K = { ð¶ ð , . . . , ð¶ ðð } and C = { ð¶ ðð + , . . . , ð¶ ðð + ð } where ð †âð + and ð †ð . The cycles ð¶ , . . . , ð¶ ð are of size ð + , that is, ð¶ ð , . . . , ð¶ ðð are ( ð + ) -cliques whereeach edge has multiplicity 2 . For ð â [ ð + ð ] , let ð ð , ð ð , . . . , ð ðð + be the ï¬rst ð + ð¶ ð and let ðŸ ð denote the clique on these ð + P ð between cliques ðŸ ð and ðŸ ð + for all ð â [ ð + ð ] , where we denote ðŸ ð + ð + = ðŸ . In [11] it is shown that in ðº there exists the ð ð¡â power of a path on vertices ð , ð , . . . , ð ð¡ with ð¡ = ð ( ð ) connecting cliques ðŸ ð and ðŸ ð + such that ð ð â ð ðð and ð ð¡ â ð â + ð â ð ð + ð for ð â [ ð + ] . Additionally, these vertices can be chosen such that | ð ( ð , ð , . . . , ð ð ) â© ð ðð + | > ( ð â ð ) ð ð and | ð ( ð ð¡ , ð ð¡ â , . . . , ð ð¡ + â ð ) â© ð ð + ð + â ð | > ( ð â ð ) ð ð, â ð â [ ð ] , (2)which ensures we can later extend this to an ð ð¡â power of a Hamilton cycle in ðº .
An important thing tonote is that these paths can be constructed one by one such that the paths P , P , . . . , P ð â only depend on K . The arguments from [11] translate directly to our case.Next we add some more vertices to the exceptional set ð . From a cluster ð ðð in a cycle ð¶ ð we move to ð all vertices ð£ not used on the paths P , . . . , P ð â for which there is a ð â² such that { ð ðð , ð ðð â² } â ðž ( ð¶ ð ) and deg ( ð£, ð ðð â² ) †ð | ð ðð â² | , where we consider the original clusters ð ðð â² , that is, we also consider the vertices already used in theconnecting paths in this calculation. Because of ( ð, ð ) -regularity and because | ð¶ ð | †ð for all ð â [ ð + ð ] by deï¬nition of a ð¶ ð -tiling, there are at most | ð¶ ð | · ð | ð ðð | †ð ðð such vertices in each cluster ð ðð . Then wemove the smallest possible number of verties to ð so that each cluster has the same number of vertices.We denote the new number of vertices per cluster by ð â² and we still write ð for the enlarged exceptionalset which now satisï¬es | ð | †ð ðð. For each vertex ð£ â ð we ï¬nd all cliques ðŸ ð with ð â [ ð â ] suchthat deg ( ð£, ð ðð ) ⥠ð | ð ðð | , â ð â [ ð + ] . Again, in the above we consider the original clusters ð ðð . Let ð¥ denote the number of such clusters. Wehave (cid:18) â ð + + ð (cid:19) ð †deg ( ð£ ) †ð + ð à ð | ð¶ ð | ! ð â² + | ð | + ð¥ ( ð + ) ð â² + ( ð â â ð¥ ) (cid:16) ðð â² + ð ð â² (cid:17) . Using ð †ð , | ð¶ ð | †ð and that â is suï¬ciently large, a simple calculation yields ð¥ ⥠ðâ / . We assign ð£ to one of these cliques such that no clique is assigned too many vertices. A greedy algorithm assignsat most ð ð â² vertices to each clique for ð = ð ðð . Now, we extend the path P ð to contain all verticesassigned to clique ðŸ ð by using additional three vertices from each of the clusters ð ðð , ð â [ ð + ] per addedvertex. This can be done without changing the ï¬rst and last ð vertices on the path so that property 2 is stillsatisï¬ed. Again, the argument in [11] applies directly to our case.8ote that the number of unused vertices in a given cluster, i.e. those not used in any of the paths, is atleast ( â ð ) ð Ⲡ⥠( â ðŒ ð ) ð. (3)We are in the same situation as in [11]: for each cycle ð¶ ð we have the same number of remaining vertices ineach ð ðð and the endpoints of the connecting paths satisfy 2, so we can apply Lemma 2.8 and the subsequentremark to close the ð ð¡â power of a Hamilton cycle ð» ð . From 3, it easily follows that ð» ð satisï¬es (i). Indeed,for every edge ( ð ð¥ , ð ðŠ ) in some ð¶ ðð , ð â [ ð + ð ] , we included in ð» ð at least mul ð¶ ð ( ð ð¥ , ð ðŠ ) · ( â ðŒ ð ) ð corresponding edges of color ð ð ( ð ð¥ , ð ðŠ ) . Apart from these, there are at most (| ð | + ðŒ ð ð ) · ð †ðŒ ð edgesin ð» ð . Hence, | ð ( ð» ð ) â ðð ð (T ) | †ðŒ ð ð | ð ð (T ) | + ðŒ ð †ðŒ ð ð · ð â + ðŒ ð †ðŒð. Now suppose we are given two tilings T , T as in the statement of the proposition. We construct two ð ð¡â powers of a Hamilton cycle ð» ð and ð» ð with a small modiï¬cation to the above procedure to ensure theydo not diï¬er too much. We think of the above algorithm as three stages of adding edges to a subgraphwhich eventually becomes the ð ð¡â power of a Hamilton cycle. We use I and I to denote the runs of thealgorithm on T = T and T = T , respectively.The ï¬rst stage of connecting the cliques is done exactly as above. Note that the paths P , . . . , P ð â are thesame in both instances. Let ð denote the exceptional set and ð the set of vertices used for the connectingpaths in I . Analogously deï¬ne ð and ð with respect to I . Note that | ð Î ð | †( ð + ) · ð ( ð ) = ð ( ð ) because the two runs only deviate after ï¬nding the ï¬rst ð â ð ( ð ) . Sincethe degree of any vertex in the ð ð¡â power of a Hamilton cycle is 2 ð, this implies the two subgraphs diï¬erin ð ( ð ) edges at this point.We can assume that the new number of vertices per cluster ð â² is the same in I and I , otherwise simplyadd a few vertices to ð ð if ð â² is larger in I ð . Set ð = ð ⪠ð and note that now | ð | †ð ðð which doesnot aï¬ect the above argument. To make the two found subgraphs similar, we treat ð \ ð and ð \ ð asthe new exceptional sets for I and I , respectively. In the second stage, we add the exceptional verticesto paths P , . . . , P ð â . As noted above, the cliques to which a vertex can be assigned do not depend on theconnecting paths. Therefore, we can assign the vertices in ð \ ( ð ⪠ð ) to the same cliques for the tworuns. We can then embed the vertices from ð \ ( ð ⪠ð ) in exactly the same way in I and I . Thus I and I only deviate after embedding all but at most | ð Î ð | = ð ( ð ) vertices. Each added vertex extendsthe path by ð ( ð ) vertices, so this stage introduces at most ð ( ð ) edges to ð» ð Î ð» ð . Finally, in the third stage, we ï¬nd connecting paths inside cliques ðŸ ð . The edges in these paths have thesame labels in I and I for all cliques ðŸ ð , ð â [ ð ] . The number of vertices left in cliques ðŸ , . . . , ðŸ ð mightdiï¬er in I and I at this point, but in total only by | ð Î ð | · ð ( ð ) . Hence, the diï¬erence between thevalues of edges in cliques ðŸ , . . . , ðŸ ð , introduced at this stage is at most ð ( ð ) . Since for each edge in C weadd ð Ⲡ⥠( â ðŒ ð ) ð edges to ð» ð and analogously for C and ð» ð , we have | ð ( ð» ð ) â ð ( ð» ð ) | ⥠(cid:16) â ðŒ ð (cid:17) ð · | ð ð (C ) â ð ð (C ) | â ð ( ð ) ⥠ð (cid:12)(cid:12) ð ð (C ) â ð ð (C ) (cid:12)(cid:12) â ð¶ð . (cid:3) It is enough to prove the theorems for ð ⪠/ ð . We deï¬ne additional constants ðŸ, ð, ðœ, ð > ð , â , ð¿ â N such that 0 < / ð ⪠ðŸ ⪠/ ð¿ †/ â ⪠ð ⪠ðœ ⪠ð = ð / ⪠/ ð, ð¿ is the constant obtained from Lemma 2.5 with parameters ð, â . Let ðº be a graph with ð ⥠ð vertices and ð : ðž ( ðº ) â {â , } an edge labelling as in the statement of the theorem.Both proofs start by applying Lemma 2.5 to ðº with parameters ð, ð and â . We thus obtain the reducedgraph ð whose vertices are clusters ð , ð , . . . , ð â , where each of the clusters ð ð is of size ð. We also havethe exceptional set ð of size at most ðð. Recall that ð + â. The reduced graph ð inherits the edgelabelling ð ð as given in 1.From Proposition 5.1, we derive two claims that will be useful for our proof. Claim 6.1.
Suppose there exists a ðŸ ð + -tiling T of ð such that | ð ð (T ) | ⥠ðœâ. Then, in ðº there exists the ð ð¡â power of a Hamilton cycle ð» ð satisfying | ð ( ð» ð ) | ⥠ðŸð. (cid:3) Claim 6.2.
Let ð¹ be a subgraph of ð on at most ð vertices and let F = { ð¶ , ð¶ , . . . , ð¶ ð } and F = { ð¶ , ð¶ , . . . , ð¶ ð } be two ð¶ ð -templates of ð¹ such that each vertex of ð¹ appears exactly ð times in F and ð times in F for some ð †ð . If F and F have diï¬erent discrepancies with respect to ð ð , then in ðº thereexists the ð ð¡â power of a Hamilton cycle ð» ð satisfying | ð ( ð» ð ) | ⥠ðŸð. Proof.
Split each of the clusters ð ð of the regular partition into ð clusters of size ð â² = â ð / ð â and put theremaining vertices in ð . Let ð ( ð ð ) denote the set of ð clusters formed by ð ð . Deï¬ne ð â² as a blow-up of ð with the edge coloring ð ð â² in the natural way: if ð ð and ð ð were joined by an edge, then put a completebipartite graph between ð ( ð ð ) and ð ( ð ð ) with all edges of color ð ð ( ð ð , ð ð ) . Let ð¹ â² denote the correspondingblow-up of ð¹ .
Formally, we deï¬ne ð ( ð¹ â² ) = à ð ð â ð ( ð¹ ) ð ( ð ð ) and ðž ( ð¹ â² ) = à ( ð ð ,ð ð ) â ðž ( ð¹ ) ð â² [ ð ( ð ð ) , ð ( ð ð )] . ByLemma 2.7, we get that all clusters joined by an edge in ð â² form an ð â² regular pair in ðº for ð â² = ðð andhave density at least ð â² = ð â ð. For the new exceptional set ð we have | ð | †ð â² ð. Note that ð â² satisï¬es ð¿ ( ð â² ) ⥠( â ð + + ð ) | ð â² | and so ð¿ ( ð â² \ ð¹ â² ) ⥠( â ð + ) | ð â² | . Applying Theorem 2.4, weobtain a ðŸ ð + -tiling K of ð â² \ ð¹ â² . Using F we construct a ð¶ ð -tiling C of ð¹ â² as follows. In the cycles in F , simply replace every occurrence of a vertex ð ð with a diï¬erent vertex from ð ( ð ð ) . It is easy to verify that C is a ð¶ ð -tiling of ð¹ â² . Analogously, we construct C from F . Let T = K . ⪠C and T = K . ⪠C . We apply Proposition 5.1 to obtain two ð ð¡â powers of Hamilton cycles ð» ð and ð» ð which satisfy (ii) with respect to ð â² and ð ð â² . Finally, note that ð ð â² (T ð ) = ð ð â² (K) + ð ð â² (C ð ) and ð ð â² (C ð ) = ð ð (F ð ) for ð = , . Hence, we have | ð ( ð» ð ) â ð ( ð» ð ) | ⥠ð â² | ð ð â² (T ) â ð ð â² (T ) | ⥠ð ( ð + ) · ⥠ðŸð. Therefore, at least one of ð» ð , ð» ð has absolute discrepancy at least ðŸð. (cid:3) First we resolve the case ð = Proof of Theorem 1.5.
Recall that ð¿ ( ðº ) ⥠( / + ð ) ð, so by Fact 2.6, we get ð¿ ( ð ) ⥠( / + ð / ) | ð | . Let ðœ bethe value of ðŸ given by Theorem 1.3 with parameters ð = ð / . Applying Theorem 1.3 to the reducedgraph ð we obtain a ðŸ -tiling T of ð of absolute discrepancy at least ðœâ. By Claim 6.1, ðº contains thesquare of a Hamilton cycle with absolute discrepancy at least ðŸð. (cid:3) In the rest of the paper, we ï¬nish the proof of Theorem 1.4.
Proof of Theorem 1.4.
We prove the theorem by contradiction, so we assume that ðº does not contain the ð ð¡â power of a Hamilton cycle with absolute discrepancy at least ðŸð. Recall that we have ð ⥠ð¿ ( ðº ) ⥠(cid:0) â ð + + ð (cid:1) ð. By Fact 2.6, the reduced graph satisï¬es ð¿ ( ð ) ⥠(cid:18) â ð + + ð (cid:19) â. (4)10is yields the following simple observation. Claim 6.3.
For any ð£ , ð£ , . . . , ð£ ð + â ð , we have | ð ( ð£ , ð£ , . . . , ð£ ð + ) | ⥠( ð + ) ðâ / . (cid:3) The following claim shows that ð is highly structured with respect to ð ð . Claim 6.4.
Let ðŸ be a clique in ð of size ð †ð + . Then ðŸ is a copy of one of the following: ðŸ + ð , ðŸ â ð , ( ðŸ ð , +) -staror ( ðŸ ð , â) -star.Proof. First we prove the claim for ( ð + ) -cliques. Let ðŸ = { ð£ , ð£ , . . . , ð£ ð + } be an ( ð + ) -clique in ð . Deï¬ne ð¶ = ( ð£ , ð£ , ð£ , ð£ , ð£ , . . . , ð£ ð + ) and ð¶ = ( ð£ , ð£ , ð£ , ð£ , ð£ , . . . , ð£ ð + ) . We can view { ð¶ } and { ð¶ } as ð¶ ð -templates on ðŸ .
Thus, by Claim 6.2, we get ð ð ( ð¶ ð ) = ð ð ( ð¶ ð ) . Note that ð ð ( ð¶ ð ) = à †ð < ð †ð + ð ð ( ð£ ð , ð£ ð ) â ð + à ð = ð ð ( ð£ ð , ð£ ð + ) , where we denote ð£ ð + = ð£ . Hence, 0 = ð ð ( ð¶ ð ) â ð ð ( ð¶ ð ) = ð ð ( ð£ , ð£ ) + ð ð ( ð£ , ð£ ) â ð ð ( ð£ , ð£ ) â ð ð ( ð£ , ð£ ) . Asthe enumeration of the vertices was arbitrary, for any distinct ð, ð, ð, ð â ðŸ, the following holds: ð ð ( ð, ð ) + ð ð ( ð, ð ) = ð ð ( ð, ð ) + ð ð ( ð, ð ) . (5)The rest of the proof appears in [2]. We present a slightly shorter argument. Assume that ðŸ is notmonochromatic, so there exists a vertex ð£ with ð + ðŸ ( ð£ ) , ð â ðŸ ( ð£ ) â â . Without loss of generality, assume | ð + ðŸ ( ð£ ) | ⥠ð¢ â ð â ðŸ ( ð£ ) . Consider arbitrary ð¥, ðŠ â ð + ðŸ ( ð£ ) . By (5), we get ð ð ( ð¥, ð£ ) + ð ð ( ð¢, ðŠ ) = ð ð ( ð¥, ðŠ )+ ð ð ( ð¢, ð£ ) . By, deï¬nition ð ð ( ð¥, ð£ ) = ð ð ( ð¢, ð£ ) = â , so this implies ð ð ( ð¢, ðŠ ) = â ð ð ( ð¥, ðŠ ) = . If | ð â ðŸ ( ð£ ) | ⥠, a completely analogous argument shows ð ð ( ð¢, ðŠ ) = , a contradiction. From this weconclude ð â ð ( ð£ ) = { ð¢ } . Applying the same reasoning to every pair ð¥, ðŠ â ð + ð ( ð£ ) , we get ð ( ð¥, ðŠ ) = ð ( ð¢, ðŠ ) = â ð¥, ðŠ â ð + ð ( ð£ ) . In other words, ðŸ is a ( ðŸ ð + , â) -star with ð¢ as its head.Now, suppose ðŸ is a clique in ð of size ð †ð + . By Claim 6.3, ðŸ can be extended, vertex by vertex, tosome clique ðŸ â² of size ð + . The statement now easily follows from the result for ( ð + ) -cliques. (cid:3) By (4), ð has large minimum degree, so we can apply Theorem 2.4 to obtain a ðŸ ð + -tiling T of ð . FromClaim 6.4 we conclude there are only four types of cliques in T . Let ðŽ denote the set of ðŸ + ð + in T ; ðµ theset of ðŸ â ð + in T ; ð¶ the set of ( ðŸ ð + , +) -stars in T ; and ð· the set of ( ðŸ ð + , â) -stars in T . Without loss ofgenerality, we may assume | ðµ | + | ð¶ | ⥠| ðŽ | + | ð· | . (6)Under the assumption that ðº does not have ð ð¡â powers of Hamilton cycles with large discrepancy, weestablish several claims about edges between cliques of diï¬erent types. We state these claims here anddefer their proofs to the end of the paper. Claim 6.5.
Consider a vertex ð¥ of a clique ð â ðŽ and let ð be a copy of ðŸ â ð + in ðµ. Then we may assume deg ( ð¥ , ð ) †ð â . Claim 6.6.
Consider a vertex ð¥ of a clique ð â ðŽ and let ð be a ( ðŸ ð + , +) -star in ð¶. Then we may assume deg ( ð¥ , ð ) †ð â . Claim 6.7.
Suppose ð¥ is the head of a clique ð â ð¶ and let ð â ð· .
Then, we may assume deg ( ð¥ , ð ) †ð â . ðº has no ð ð¡â power of a Hamilton cycle with absolute discrepancy at least ðŸð. We can assume | ð ð (T ) | < ðœâ, asotherwise we can ï¬nd the desired ð ð¡â power of a Hamilton cycle by Claim 6.1.First, we show that ðŽ = â . Otherwise, consider some vertex ð£ of a clique in ðŽ. By Claims 6.5 and 6.6, ð£ canhave at most ð â ðµ ⪠ð¶. Trivially, it can have at most ð + ðŽ ⪠ð· .
Clearly, | ðŽ | + | ðµ | + | ð¶ | + | ð· | = â /( ð + ) . Using (6), we getdeg ( ð£ ) †( ð â ) (| ðµ | + | ð¶ |) + ( ð + ) (| ðŽ | + | ð· |) †ðð + â, which contradicts the degree assumption (4). Hence, ðŽ = â . Note that | ðµ | + | ð¶ | â | ð· | †ðœâ, as otherwise | ð ð (T ) | ⥠ðœâ. Hence, we have 2 (| ðµ | + | ð¶ |) †(cid:0) ð + + ðœ (cid:1) â. Suppose ð¶ is nonempty and let ð£ be the head of some ( ðŸ ð + , +) -star in ð¶. By Claim 6.7, it can have at most ð â ð· .
Therefore,deg ( ð£ ) †( ð â ) | ð· | + ( ð + ) (| ðµ | + | ð¶ |) = ( ð â ) âð + + (| ðµ | + | ð¶ |) †(cid:18) â ð + + ðœ (cid:19) â. This contradicts (4), so ð¶ = â . Finally, from (6), we have | ðµ | ⥠| ð· | which yields ð ð (T ) = â (cid:18) ð + (cid:19) | ðµ | + (cid:18) (cid:18) ð + (cid:19) â ð (cid:19) | ð· | †â ðœâ, a contradiction. (cid:3) Proofs of Claims 6.5, 6.6 and 6.7
In the following we will consider two ( ð + ) -cliques in T which we denote by ð = { ð¥ , . . . , ð¥ ð + } and ð = { ðŠ , . . . , ðŠ ð + } . Additionally, we assume that ð¥ has at least ð edges towards vertices of ð .
Without lossof generality, we assume ð¥ is connected to ðŠ , ðŠ , . . . , ðŠ ð . From Claim 6.3 it follows that there is a vertex ð¥ â² such that ð â² = ð ⪠{ ð¥ â² } forms an ( ð + ) -clique. Similarly, there is a vertex ðŠ â² such that ð â² = ð ⪠{ ðŠ â² } formsan ( ð + ) -clique and all the vertices ð¥ , . . . ð¥ ð + , ð¥ â² , ðŠ , . . . , ðŠ ð + , ðŠ â² are distinct. We construct two templateson ð¹ = ð â² . ⪠ð â² and show, for the cases mentioned in the claims, that these templates have diï¬erentdiscrepancy. By Claim 6.2, this contradicts the assumption that ðº has no ð ð¡â power of a Hamilton cyclewith a large discrepancy. We start by deï¬ning four cycles which will be used in the templates. ð¶ = ( ð¥ , ð¥ , . . . , ð¥ ð + , ð¥ â² ) ð¶ = ( ð¥ , ð¥ , ð¥ , . . . , ð¥ ð + , ð¥ â² ) ð¶ = ( ðŠ , ðŠ , ðŠ , . . . , ðŠ ð + , ðŠ â² ) ð¶ = ( ð¥ , ðŠ , ðŠ , . . . , ðŠ ð ,ðŠ ð + , ðŠ â² , ðŠ , ðŠ , . . . , ðŠ ð â , ðŠ ð â ,ðŠ ð + , ðŠ â² , ðŠ , ðŠ , . . . , ðŠ ð â , ðŠ ð ,. . . ,ðŠ ð + , ðŠ â² , ðŠ , ðŠ , . . . , ðŠ ð ,ðŠ ð + , ðŠ â² , ðŠ , ðŠ , . . . , ðŠ ð ,ðŠ ð + , ðŠ â² , ðŠ , ðŠ , . . . , ðŠ ð ) Using these, we deï¬ne two templates F and F as follows: F = (cid:0) ( ð + ) à ð¶ , ( ð + ) à ð¶ (cid:1) and F = (cid:0) ð¶ , ð à ð¶ , ð¶ (cid:1) , ð à ð¶ ð to indicate ð copies of ð¶ ð . Note that F contains each vertex in ð¹ exactly ð + ð â² \ { ð¥ } appears once in ð¶ and ð¶ , so F contains each of these vertices ð + ð¶ contains ð¥ once and each of the vertices in ð â² exactly ð + | ð¶ | = + ( ð + ) ( ð + ) = ð + ð + , (7)which will be useful later on. Additionally, it is easy to see that F contains each vertex in ð¹ exactly ð + F and F have diï¬erent discrepancies, we reach a contradiction by Claim 6.2. In thefollowing claims we show this is true for several cases of interest. When calculating the discrepancy ofa particular ð ð¡â power of a cycle, we will mostly use the following principle. As the ( ð + ) -cliques weconsider are highly structured, most of the edge values under consideration are known given the types ofcliques of ð and ð .
More precisely, we ï¬nd a small subset of edges ðž â² such that all edges in ðž ( ð¶ ðð ) \ ðž â² havethe same color ð . Then, we can calculate the discrepancy of ð¶ ðð as ð ð ( ð¶ ðð ) = ð (cid:0) ð | ð¶ ð | â | ðž â² | (cid:1) + ð ð ( ðž â² ) . Additionally, observe that ð ð (F ) â ð ð (F ) = â ð ð ( ð¶ ð ) + ð ð ( ð¶ ð ) + ( ð + ) ð ð ( ð¶ ð ) â ð ð ( ð¶ ð ) . (8)Finally, we proceed to prove the individual claims. Claim 6.5.
Consider a vertex ð¥ of a clique ð â ðŽ and let ð be a copy of ðŸ â ð + in ðµ. Then we may assume deg ( ð¥ , ð ) †ð â . Proof.
Suppose deg ( ð¥ , ð ) ⥠ð and let F , F be deï¬ned as above. Applying Claim 6.4 to the clique ð â² , weget that all edges from ð¥ â² to ð have the same color ð ð ( ð¥ â² , ð¥ ) . Analogously, all edges from ðŠ â² to ð havecolor ð ð ( ðŠ â² , ðŠ ) and all edges from ð¥ to ð \ { ðŠ ð + } have color ð ð ( ð¥ , ðŠ ) . We calculate: ð ð ( ð¶ ð ) = ( ð + ) ð â ð + ð ð ð ( ð¥ â² , ð¥ ) = ð â ð + ð ð ð ( ð¥ â² , ð¥ ) ð ð ( ð¶ ð ) = ( ð + ) ð â ð + ð ð ð ( ð¥ â² , ð¥ ) = ð + ð ð ð ( ð¥ â² , ð¥ ) ð ð ( ð¶ ð ) = â( ( ð + ) ð â ð ) + ð ð ð ( ðŠ â² , ðŠ ) = â ð + ð ð ð ( ðŠ â² , ðŠ ) ð ð ( ð¶ ð ) = â (cid:0) ð | ð¶ | â ð â ð ( ð + ) (cid:1) + ð ð ð ( ð¥ , ðŠ ) + ð ( ð + ) ð ð ( ðŠ â² , ðŠ ) = â ð â ð + ð + ð ð ð ( ð¥ , ðŠ ) + ð ( ð + ) ð ð ( ðŠ â² , ðŠ ) . Plugging these values into (8), we obtain: ð ð (F ) â ð ð (F ) = â ð ð ð ( ð¥ , ðŠ ) â . We are done by Claim 6.2. (cid:3)
Claim 6.6.
Consider a vertex ð¥ of a clique ð â ðŽ and let ð be a ( ðŸ ð + , +) -star in ð¶. Then we may assume deg ( ð¥ , ð ) †ð â . Proof.
Assume deg ( ð¥ , ð ) ⥠ð and deï¬ne F , F as above. By Claim 6.4, all edges from ð¥ â² to ð have thesame color ð ð ( ð¥ â² , ð¥ ) . By Claim 6.4, ð â² is a ( ðŸ ð + , +) -star with its head in ð .
Note that the values of ð ð ( ð¶ ð ) and ð ð ( ð¶ ð ) are as in the previous claim; we also calculate ð ð ( ð¶ ð ) : ð ð ( ð¶ ð ) = ð â ð + ð ð ð ( ð¥ â² , ð¥ ) ð ð ( ð¶ ð ) = ð + ð ð ð ( ð¥ â² , ð¥ ) ð ð ( ð¶ ð ) = â (cid:0) ( ð + ) ð â ð (cid:1) + ð = â ð ( ð â ) . Now, we consider two cases:(a) ðŠ ð + is the head of ð .
Applying Claim 6.4 to ( ð \ { ðŠ ð + }) ⪠{ ð¥ } , we conclude that all edges from ð¥ to ðŠ ð , ð â [ ð ] are ofcolor ð ð ( ð¥ , ðŠ ) . Using this, we obtain: ð ð ( ð¶ ð ) = â( ð | ð¶ | â ð â ð ( ð + )) + ð ð ð ( ð¥ , ðŠ ) + ð ( ð + ) â ð + ð + ð + ð ð ð ( ð¥ , ðŠ ) . Substituting into (8), we have ð ð (F ) â ð ð (F ) = â ð ð ð ( ð¥ , ðŠ ) â . (b) ðŠ ð + is not the head of ð .
Applying Claim 6.4 to ( ð \ { ðŠ ð + }) ⪠{ ð¥ } , we get that all edges in ð¶ ð incident to the head of ð havevalue + , while all other edges have value â . From this we have: ð ð ( ð¶ ð ) = â( ð | ð¶ | â ð ( ð + )) + ð ( ð + ) = â ð + ð + ð and ð ð (F ) â ð ð (F ) = ð â (cid:3) Claim 6.7.
Suppose ð¥ is the head of a clique ð â ð¶ and let ð â ð· .
Then, we may assume deg ( ð¥ , ð ) †ð â . Proof.
Again, suppose deg ( ð¥ , ð ) ⥠ð and deï¬ne F and F as above. By Claim 6.4, ð â² is a ( ðŸ ð + , â) -starwith its head in ð .
We have ð ð ( ð¶ ð ) = â ð ( ð + ) , ð ð ( ð¶ ð ) = â ð ( ð â ) and ð ð ( ð¶ ð ) = ð ( ð â ) . Note that the same edges of ð¶ ð are known as those in Claim 6.6, but are of opposite value. Again, weconsider two cases:(a) ðŠ ð + is the head of ð .
Applying Claim 6.4 to ( ð \ { ðŠ ð + }) ⪠{ ð¥ } , we conclude that all edges from ð¥ to ðŠ ð , ð â [ ð ] are ofcolor ð ð ( ð¥ , ðŠ ) . So, we get ð ð ( ð¶ ð ) = ð â ð â ð + ð ð ð ( ð¥ , ðŠ ) . Substituting into (8), we have ð ð (F ) â ð ð (F ) = ð â ð ð ð ( ð¥ , ðŠ ) â . (b) ðŠ ð + is not the head of ð .
We apply Claim 6.4 to ( ð \ { ðŠ ð + }) ⪠{ ð¥ } and obtain that all edges in ð¶ ð incident to the head of ð have value â , while all other edges have value + . From this we have: ð ð ( ð¶ ð ) = ð â ð â ð and ð ð (F ) â ð ð (F ) = ð â (cid:3) Acknowledgments
The author would like to thank MiloËs TrujiÂŽc for introducing him to the problem, helpful discussions andmany useful comments which helped improve the presentation of this paper.
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