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 Bradaหc โ March 1, 2021
Abstract
The Pยดosa-Seymour conjecture asserts that every graph on ๐ vertices with minimum degree at least ( โ /( ๐ + )) ๐ contains the ๐ ๐กโ power of a Hamilton cycle. Komlยดos, Sยดarkยจozy and Szemerยดedi famouslyproved the conjecture for large ๐. The notion of discrepancy appears in many areas of mathematics,including graph theory. In this setting, a graph ๐บ is given along with a 2-coloring of its edges. Oneis then asked to ๏ฌnd in ๐บ a copy of a given subgraph with a large discrepancy, i.e., with many moreedges in one of the colors. For ๐ โฅ , we determine the minimum degree threshold needed to ๏ฌndthe ๐ ๐กโ power of a Hamilton cycle of large discrepancy, answering a question posed by Balogh, Csaba,Pluhยดar and Treglown. Notably, for ๐ โฅ , this threshold approximately matches the minimum degreerequirement of the Pยดosa-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 ๏ฌrst signi๏ฌcant 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 ๏ฌelds such as ergodic theory, number theory,statistics, geometry, computer science, etc. For a comprehensive overview of the ๏ฌeld, 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 ade๏ฌnition. De๏ฌnition 1.1.
Let ๐บ = ( ๐ , ๐ธ ) be a graph and ๐ : ๐ธ โ {โ , } a labelling of its edges. Given a subgraph ๐ป of ๐บ, we de๏ฌne its discrepancy ๐ ( ๐ป ) as ๐ ( ๐ป ) = ร ๐ โ ๐ธ ( ๐ป ) ๐ ( ๐ ) . Furthermore, we refer to the value | ๐ ( ๐ป ) | as the absolute discrepancy of ๐ป .
One of the central questions in graph discrepancy theory is the following. Suppose we are given a graph ๐บ and a spanning subgraph ๐ป .
Does ๐บ, for every edge labelling ๐ : ๐ธ ( ๐บ ) โ {โ , } , contain an isomorphiccopy of ๐ป of high absolute discrepancy with respect to ๐ ? Erdหos, Fยจuredi, Loebl and Sยดos [6] proved the ๏ฌrstresult of this kind. They show that, given a tree on ๐ vertices ๐ ๐ with maximum degree ฮ and a {โ , } -coloring of the edges of the complete graph ๐พ ๐ , one can ๏ฌnd a copy of ๐ ๐ with absolute discrepancy at least ๐ ( ๐ โ โ ฮ ) , for some absolute constant ๐ > . โ Department of Computer Science, ETH Zยจurich, 8092 Zยจurich, Switzerland. Email: [email protected] commonly studied topic in extremal combinatorics are
Dirac-type problems where one is given a graph ๐บ on ๐ vertices with minimum degree at least ๐ผ๐ and wants to prove that ๐บ contains a copy of a speci๏ฌcspanning subgraph ๐ป . In the discrepancy setting it is natural to ask whether we can also ๏ฌnd a copy of ๐ป with large absolute discrepancy. Balogh, Csaba, Jing and Pluhยดar 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 Pluhยดar [1]) . Let < ๐ < / and ๐ โ N be su๏ฌciently large. If ๐บ isan ๐ -vertex graph with ๐ฟ ( ๐บ ) โฅ ( / + ๐ ) ๐ and ๐ : ๐ธ ( ๐บ ) โ {โ , } , then ๐บ contains a Hamilton cycle with absolute discrepancy at least ๐๐ / withrespect to ๐ . Moreover, if divides ๐, there is an ๐ -vertex graph with ๐ฟ ( ๐บ ) = ๐ / and an edge labelling ๐ : ๐ธ ( ๐บ ) โ {โ , } such that any Hamilton cycle in ๐บ has discrepancy with respect to ๐ . 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-Szemerยดedi theorem. It states that if ๐ divides ๐ and ๐บ is a graph on ๐ vertices with ๐ฟ ( ๐บ ) โฅ ( โ / ๐ ) ๐, then ๐บ contains a perfect ๐พ ๐ -tiling, i.e. its vertexset can be partitioned into disjoint cliques of size ๐ . Balogh, Csaba, Pluhยดar and Treglown [2] proved adiscrepancy version of this theorem.
Theorem 1.3 (Balogh, Csaba, Pluhยดar and Treglown [2]) . Suppose ๐ โฅ is an integer and let ๐ > . Thenthere exists ๐ โ N and ๐พ > such that the following holds. Let ๐บ be a graph on ๐ โฅ ๐ vertices where ๐ divides ๐ and where ๐ฟ ( ๐บ ) โฅ (cid:18) โ ๐ + + ๐ (cid:19) ๐. Given any function ๐ : ๐ธ ( ๐บ ) โ {โ , } there exists a perfect ๐พ ๐ -tiling T in ๐บ so that (cid:12)(cid:12)(cid:12) ร ๐ โ ๐ธ (T) ๐ ( ๐ ) (cid:12)(cid:12)(cid:12) โฅ ๐พ๐. The ๐ ๐กโ power of a graph ๐บ is the graph on the same vertex set in which two vertices are joined by an edge ifand only if their distance in ๐บ is at most ๐ . The Pยดosa-Seymour conjecture asserts that any graph on ๐ verticeswith minimum degree at least ( โ /( ๐ + )) ๐ contains the ๐ ๐กโ power of a Hamilton cycle. Komlยดos, Sยดarkยจozyand Szemerยดedi [12] proved the conjecture for large ๐. In [2] the authors posed the question of determiningthe minimum degree needed to force the ๐ ๐กโ power of a Hamilton cycle with absolute discrepancy linearin ๐. Because the ๐ ๐กโ power of a Hamilton cycle contains a (almost) perfect ( ๐ + ) -tiling, they suggestedthe minimum degree required should be ( โ /( ๐ + ) + ๐ ) ๐, based on their result for ๐พ ๐ -tilings. We provethis value is correct for ๐ = . However, we show that for ๐ โฅ , a minimum degree of ( โ /( ๐ + ) + ๐ ) ๐, for arbitrarily small ๐ > , is su๏ฌcient, approximately matching the minimum degree required for ๏ฌndingany ๐ ๐กโ power of a Hamilton cycle. As far as the author knows, this is the ๏ฌrst Dirac-type discrepancyresult in which the threshold for ๏ฌnding a spanning subgraph of large discrepancy is the same, up to anarbitrarily small linear term, as the minimum degree required for ๏ฌnding any copy of the subgraph. Theorem 1.4.
For any integer ๐ โฅ and ๐ > , there exist ๐ โ N and ๐พ > such that the following holds.Suppose a graph ๐บ on ๐ โฅ ๐ vertices with minimum degree ๐ฟ ( ๐บ ) โฅ ( โ /( ๐ + ) + ๐ ) ๐ and an edge coloring ๐ : ๐ธ ( ๐บ ) โ {โ , } are given. Then in ๐บ there exists the ๐ ๐กโ power of a Hamilton cycle ๐ป ๐ satisfying (cid:12)(cid:12)(cid:12) ร ๐ โ ๐ธ ( ๐ป ๐ ) ๐ ( ๐ ) (cid:12)(cid:12)(cid:12) โฅ ๐พ๐. ๐ ๐กโ power of a Hamilton cycle of large discrepancyis the same for ๐ โ { , , } and equals ( + ๐ ) ๐. The cases ๐ = , ๐ = . Theorem 1.5.
For any ๐ > , there exist ๐ โ N and ๐พ > such that the following holds. Suppose a graph ๐บ on ๐ โฅ ๐ vertices with minimum degree ๐ฟ ( ๐บ ) โฅ ( / + ๐ ) ๐ and an edge coloring ๐ : ๐ธ ( ๐บ ) โ {โ , } aregiven. Then in ๐บ there exists the square of a Hamilton cycle ๐ป satisfying (cid:12)(cid:12)(cid:12) ร ๐ โ ๐ธ ( ๐ป ) ๐ ( ๐ ) (cid:12)(cid:12)(cid:12) โฅ ๐พ๐. These results are tight in the following sense. If we weaken the minimum degree requirement by replacingthe term ๐๐ with a sublinear term, then there are examples in which any ๐ ๐กโ power of a Hamilton cyclehas absolute discrepancy ๐ ( ๐ ) . The paper is organised as follows. In Section 2 we introduce some notation and de๏ฌnitions 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 byKomlยดos, Sยดarkยจozy and Szemerยดedi of an approximate version of the Pยดosa-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 ๐บ be agraph. We use ๐ ๐บ ( ๐ฃ ) to denote the neighbourhood of ๐ฃ in ๐บ .
Given an edge labelling ๐ : ๐ธ ( ๐บ ) โ {โ , } , we use ๐บ + to denote the graph containing all edges labelled 1 and ๐บ โ to denote the graph containing alledges labelled โ . We write ๐ + ๐บ ( ๐ฃ ) for the set of ๐ข in ๐ ๐บ ( ๐ฃ ) such that ๐ ( ๐ฃ, ๐ข ) = ๐ โ ๐บ ( ๐ฃ ) for the set of ๐ข in ๐ ๐บ ( ๐ฃ ) such that ๐ ( ๐ฃ, ๐ข ) = โ . For a vertex ๐ฃ and a subset of vertices ๐ , we de๏ฌne ๐ ( ๐ฃ, ๐ ) = ๐ ( ๐ฃ ) โฉ ๐ and deg ( ๐ฃ, ๐ ) = | ๐ ( ๐ฃ, ๐ ) | . We write . โช 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 de๏ฌne the multiplicity mul ๐ถ ( ๐ฃ ) of a vertex ๐ฃ in a cycle ๐ถ as the number of occurences of ๐ฃ in ๐ถ whenviewed as a closed walk. Given a cycle ๐ถ we de๏ฌne its ๐ ๐กโ power as the multigraph obtained by connectingvertices at every pair of indices at most ๐ apart in the cyclic order de๏ฌning ๐ถ. More formally, for a cycle ๐ถ = ( ๐ฃ , ๐ฃ , . . . , ๐ฃ ๐ ) , we de๏ฌne its ๐ ๐กโ power, denoted by ๐ถ ๐ = ( ๐ฃ , ๐ฃ , . . . , ๐ฃ ๐ ) ๐ , as the multigraph with thefollowing edge multiplicities:mul ๐ถ ๐ ( ๐ฅ๐ฆ ) = (cid:12)(cid:12)(cid:8) ( ๐, ๐ ) | ๐ โ [ ๐ ] , ๐ โ [ ๐ ] , { ๐ฃ ๐ , ๐ฃ ๐ + ๐ } = { ๐ฅ, ๐ฆ } (cid:9)(cid:12)(cid:12) , where we denote ๐ฃ ๐ + ๐ = ๐ฃ ๐ for 1 โค ๐ โค ๐ . The ๐ ๐กโ power of a simple ( ๐ + ) -cycle will sometimes be referredto as an ( ๐ + ) -clique. Importantly, though, it has two copies of each edge. Given an edge labelling ๐ : ๐ธ ( ๐บ ) โ {โ , } , we de๏ฌne the discrepancy of ๐ถ ๐ in the natural way: ๐ ( ๐ถ ๐ ) = ร ๐ โ ๐ธ ( ๐ถ ๐ ) mul ๐ถ ๐ ( ๐ ) ๐ ( ๐ ) . Similarly as in [2], we de๏ฌne a ๐ถ ๐ -template. Note that in the following de๏ฌnition we only allow shortcycles. De๏ฌnition 2.1.
Let ๐น be a graph. A ๐ถ ๐ -template of ๐น is a collection F = { ๐ถ , ๐ถ , . . . , ๐ถ ๐ } of not necessarilydistinct cycles whose ๐ ๐กโ powers are subgraphs of ๐น .
In a ๐ถ ๐ -template each vertex appears the same numberof times, that is, ร ๐ ๐ = mul ๐ถ ๐ ( ๐ฃ ) is the same for all ๐ฃ โ ๐ ( ๐น ) . Moreover, we require that each cycle ๐ถ ๐ haslength between ๐ + ๐ . The discrepancy of a ๐ถ ๐ -template is given as ๐ (F ) = ร ๐ ๐ = ๐ ( ๐ถ ๐๐ ) .
3e notion of a ๐ถ ๐ -tiling is obtained by adding the natural restriction that each vertex appears exactlyonce. De๏ฌnition 2.2.
Let ๐น be a graph. A ๐ถ ๐ -tiling T of ๐น is a collection of simple cycles T = { ๐ถ , ๐ถ , . . . , ๐ถ ๐ } whose ๐ ๐กโ powers are subgraphs of ๐น and each vertex appears precisely once in these cycles. The lengthof each cycle is between ๐ + ๐ . The discrepancy of a ๐ถ ๐ -tiling is given as ๐ (T ) = ร ๐ ๐ = ๐ ( ๐ถ ๐๐ ) . In the above two de๏ฌnitions, 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 ๐ ๐กโ power of a cycle. A ๐พ ๐ + -tiling is a ๐ถ ๐ -tiling in which all cycles have length ๐ + , that is, all tiles are ( ๐ + ) -cliques.We give names to special types of ๐ -cliques with respect to an edge labelling ๐ . De๏ฌnition 2.3.
We write ๐พ + ๐ for the ๐ -clique with all edges labelled 1 and ๐พ โ ๐ for the ๐ -clique with alledges labelled โ . The ( ๐พ ๐ , +) -star is the clique whose edges labelled 1 induce a copy of ๐พ ,๐ โ . The root ofthis ๐พ ,๐ โ is called the head of the ( ๐พ ๐ , +) -star. We de๏ฌne the ( ๐พ ๐ , โ) -star and its head analogously.We write ๐ผ โช ๐ฝ โช ๐พ, if the constants can be chosen from right to left such that all calculations in our proofare valid. More precisely, ๐ผ โช ๐ฝ means there is a positive increasing function ๐ ( ๐ฝ ) such that for ๐ผ = ๐ ( ๐ฝ ) , all calculations in the proof are valid. This notion naturally extends to hierarchies of larger length as well.We omit ๏ฌoors and ceilings whenever they do not a๏ฌect the argument.In our proofs we use the famous Hajnal-Szemerยดedi theorem in the following form. Theorem 2.4 (Hajnal and Szemerยดedi [9]) . Every graph ๐บ whose order ๐ is divisible by ๐ and has minimumdegree at least ( โ / ๐ ) ๐ contains a perfect ๐พ ๐ -tiling. In the proof of the main result, we use a multicolored variant of Szemerยดediโs regularity lemma [14]. Beforestating the result, we de๏ฌne the relevant notions. The density of a bipartite graph ๐บ with vertex classes ๐ด and ๐ต is de๏ฌned as ๐ ( ๐ด, ๐ต ) = ๐ ( ๐ด, ๐ต )| ๐ด || ๐ต | . Given ๐, ๐ > , the graph ๐บ is said to be ( ๐, ๐ ) -regular if ๐ ( ๐ด, ๐ต ) โฅ ๐ and for any ๐ โ ๐ด, ๐ โ ๐ต such that | ๐ | > ๐ | ๐ด | and | ๐ | > ๐ | ๐ต | , we have | ๐ ( ๐ด, ๐ต ) โ ๐ ( ๐, ๐ ) | < ๐. The graph ๐บ is ( ๐, ๐ฟ ) -super-regular if for every ๐ โ ๐ด with | ๐ | > ๐ | ๐ด | and ๐ โ ๐ต with | ๐ | > ๐ | ๐ต | , we have ๐ ( ๐, ๐ ) > ๐ฟ, and furthermore, deg ( ๐ ) > ๐ฟ | ๐ต | for all ๐ โ ๐ด and deg ( ๐ ) > ๐ฟ | ๐ด | for all ๐ โ ๐ต. Suppose we are given a graph ๐บ with an edge labelling ๐ : ๐ธ ( ๐บ ) โ {โ , } . Given disjoint
๐, ๐ โ ๐ ( ๐บ ) we write ( ๐, ๐ ) + ๐บ or ๐บ + [ ๐, ๐ ] for the bipartite graphwith vertex classes ๐, ๐ containing edges between ๐ and ๐ labelled 1 . Analogously, we de๏ฌne ( ๐, ๐ ) โ ๐บ and ๐บ โ [ ๐, ๐ ] with respect to edges labelled โ . We use a variant of the regularity lemma which is easily deduced from the multicolored version in [13].
Lemma 2.5.
For every ๐ > and ๐, โ โ N there exists ๐ฟ = ๐ฟ ( ๐, โ ) such that the following holds. Let ๐ โ [ , ) and let ๐บ be a graph on ๐ โฅ ๐ฟ vertices with an edge coloring ๐ : ๐ธ ( ๐บ ) โ {โ , } . Then, there existsa partition ( ๐ ๐ ) โ๐ = , for some โ โ [ โ , ๐ฟ ] divisible by ๐, of ๐ ( ๐บ ) and a spanning subgraph ๐บ โฒ of ๐บ with thefollowing properties:(i) | ๐ | โค ๐๐ and | ๐ | = | ๐ | = ยท ยท ยท = | ๐ โ | , ii) deg ๐บ โฒ ( ๐ฃ ) โฅ deg ๐บ ( ๐ฃ ) โ ( ๐ + ๐ ) ๐ for every ๐ฃ โ ๐ ( ๐บ ) ; (iii) ๐ ( ๐บ โฒ [ ๐ ๐ ]) = for all โค ๐ โค โ ; (iv) for all โค ๐ < ๐ โค โ and ๐ โ {+ , โ} , either ( ๐ ๐ , ๐ ๐ ) ๐๐บ โฒ is an ( ๐, ๐ ) -regular pair or ๐บ โฒ ๐ [ ๐ ๐ , ๐ ๐ ] is empty; We call ๐ , . . . , ๐ โ clusters and ๐ the exceptional set . We refer to ๐บ โฒ as the pure graph . We de๏ฌne the reducedgraph ๐ of ๐บ with parameters ๐, ๐, ๐, โ to be the graph whose vertices are ๐ , . . . , ๐ โ and where ( ๐ ๐ , ๐ ๐ ) isan edge if at least one of ( ๐ ๐ , ๐ ๐ ) + ๐บ โฒ and ( ๐ ๐ , ๐ ๐ ) โ ๐บ โฒ is ( ๐, ๐ ) -regular. On the reduced graph ๐ , we de๏ฌne theedge coloring ๐ ๐ : ๐ธ ( ๐ ) โ {โ , } as follows: ๐ ๐ ( ๐ ๐ , ๐ ๐ ) = ( , if ( ๐ ๐ , ๐ ๐ ) + ๐บ โฒ is ( ๐, ๐ ) -regular โ , otherwise. (1)Note that if both ( ๐ ๐ , ๐ ๐ ) + ๐บ โฒ and ( ๐ ๐ , ๐ ๐ ) โ ๐บ โฒ are ( ๐, ๐ ) -regular, ๐ ๐ only records the former property.We use a well-known fact about the reduced graph: Fact 2.6.
Let ๐ > be a given constant and ๐บ a graph on ๐ vertices such that ๐ฟ ( ๐บ ) โฅ ๐๐. Let ๐ be the reducedgraph obtained after applying Lemma 2.5 with parameters ๐, ๐ and โ . Then ๐ฟ ( ๐ ) โฅ ( ๐ โ ๐ โ ๐ ) | ๐ | . 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 ( ๐ด, ๐ต ) be an ๐ -regular pair with density ๐, and, for some ๐ผ > ๐, let ๐ด โฒ โ ๐ด, | ๐ด | โฒ โฅ ๐ผ | ๐ด | , ๐ต โฒ โ ๐ต, | ๐ต โฒ | โฅ ๐ผ | ๐ต | . Then ( ๐ด โฒ , ๐ต โฒ ) is an ๐ โฒ -regular pair with ๐ โฒ = ๐๐๐ฅ { ๐ / ๐ผ, ๐ } and forits density ๐ โฒ we have | ๐ โฒ โ ๐ | < ๐. We also need the incredibly useful result of Komlยดos, Sยดarkยจozy and Szemerยดedi, known as the
Blow-up lemma,which states that ( ๐, ๐ ) -superregular pairs behave like complete bipartite graphs in terms of containgsubgraphs of bounded degree. Lemma 2.8 (Blow-up lemma [10]) . Given a graph ๐ of order ๐ and positive parameters ๐ฟ, ฮ , there exists apositive ๐ = ๐ ( ๐ฟ, ฮ , ๐ ) such that the following holds. Let ๐ , ๐ , . . . , ๐ ๐ be arbitrary positive integers and let usreplace the vertices ๐ฃ , ๐ฃ , . . . , ๐ฃ ๐ of ๐ with pairwise disjoint sets ๐ , ๐ , . . . , ๐ ๐ of sizes ๐ , ๐ , . . . , ๐ ๐ (blowingup). We construct two graphs on the same vertex set ๐ = โช ๐ ๐ . The ๏ฌrst graph ๐น is obtained by replacing eachedge { ๐ฃ ๐ , ๐ฃ ๐ } of ๐ with the complete bipartite graph between the corresponding vertex-sets ๐ ๐ and ๐ ๐ . A sparsergraph ๐บ is constructed by replacing each edge { ๐ฃ ๐ , ๐ฃ ๐ } arbitrarily with an ( ๐, ๐ฟ ) -super-regular pair between ๐ ๐ and ๐ ๐ . If a graph ๐ป with ฮ ( ๐ป ) โค ฮ is embeddable into ๐น then it is already embeddable into ๐บ .
The following remark also appears in [10].
Remark.
When using the Blow-up Lemma, we usually need the following strengthened version: Given ๐ > , there are positive numbers ๐ = ๐ ( ๐ฟ, ฮ , ๐, ๐ ) and ๐ผ = ๐ผ ( ๐ฟ, ฮ , ๐, ๐ ) such that the Blow-up Lemma in the equalsize case (all | ๐ ๐ | are the same) remains true if for every ๐ there are certain vertices ๐ฅ to be embedded into ๐ ๐ whose images are a priori restricted to certain sets ๐ถ ๐ฅ โ ๐ ๐ provided that(i) each ๐ถ ๐ฅ within a ๐ ๐ is of size at least ๐ | ๐ ๐ | , (ii) the number of such restrictions within a ๐ ๐ is not more than ๐ผ | ๐ ๐ | . We present simple lower bound constructions showing our results are best possible in a certain sense. For ๐ = ๐ ( ๐ ) = ๐ ( ) , the condition ๐ฟ ( ๐บ ) โฅ ( โ ๐ + + ๐ ) ๐ when ๐ โฅ ๐ฟ ( ๐บ ) โฅ ( / + ๐ ) ๐ when ๐ =
2, is not5nough to guarantee an ๐ ๐กโ power of a Hamilton cycle with absolute discrepancy linear in ๐. Moreover,for ๐ = , there exists a graph in which every ๐ ๐กโ power of a Hamilton cycle has discrepancy 0 . First consider ๐ โฅ . We construct a graph ๐บ as follows. Let ๐ก be even and ๐ , . . . , ๐ ๐ + disjoint clusters ofsize ๐ก . Additionally, let ๐ be a cluster of size ๐. We construct a graph on the vertex set ๐ = . ร ๐ + ๐ = ๐ ๐ . Weput an edge between any two vertices from di๏ฌerent clusters and we put all edges connecting two verticesin ๐ . Let ๐ = | ๐ | = ( ๐ + ) ๐ก + ๐ and note that ๐ฟ ( ๐บ ) = ๐๐ก + ๐ = (cid:16) โ ๐ + + ๐ ( ๐ + ) (( ๐ + ) ๐ก + ๐ ) (cid:17) ๐. Next we describe the coloring ๐ of the edges. We color the edges incident to vertices in ๐ arbitrarily. Foreach ๐ ๐ , ๐ โฅ positive and the other half as negative . For a vertex ๐ฃ โ ๐ ๐ and any vertex ๐ข โ ๐ ๐ where 1 โค ๐ < ๐ we set ๐ ( ๐ฃ, ๐ข ) = ๐ฃ is positive and ๐ ( ๐ฃ, ๐ข ) = โ ๐ฃ is negative.Let ๐ป ๐ be an ๐ ๐กโ power of a Hamilton cycle in ๐บ viewed as a 2 ๐ -regular subgraph of ๐บ .
Call a vertex ๐ฃ โ ๐ \ ๐ a bad vertex if at least one of its neighbours in ๐ป ๐ is from the cluster ๐ , otherwise call it good . If a vertex ๐ฃ โ ๐ ๐ is good then in ๐ป ๐ it has precisely two neighbours from each of the clusters ๐ ๐ , โค ๐ โค ๐ + , ๐ โ ๐. Note that for ๐ โฅ ๐ ๐ can be adjacent to a vertex ๐ฃ โ ๐ , so there are at most 2 ๐ bad vertices in ๐ ๐ . Now consider only positive good vertices and their edges towards vertices from clusterswith a smaller index. Thus, the number of edges labelled 1 in ๐ป ๐ is at least ๐ + ร ๐ = ( ๐ โ ) ( ๐ก / โ ๐ ) = ๐ ( ๐ + ) ( ๐ก / โ ๐ ) . Hence, we have ๐ ( ๐ป ๐ ) โฅ โ ๐๐ + ๐ ( ๐ + ) ( ๐ก / โ ๐ ) โฅ โ ๐ ( ๐ + ) ๐. Completely analogously, ๐ ( ๐ป ๐ ) โค ๐ ( ๐ + ) ๐. Therefore, if ๐ = , we have ๐ ( ๐ป ๐ ) = ๐ = ๐ ( ๐ ) , weget | ๐ ( ๐ป ๐ ) | = ๐ ( ๐ ) . For ๐ = , the following construction was given in [1], where the case ๐ = ๐บ be the4-partite Turยดan graph on ๐ = ๐ vertices, so ๐ฟ ( ๐บ ) = ๐. Colour all edges incident to one of the parts with โ . Any square of a Hamilton cycle contains 4 ๐ edges labelled โ
1, exactly 4 for eachvertex in the special class. As it has a total of 8 ๐ edges, its discrepancy is 0 . Similarly as above, we can add ๐ = ๐ ( ๐ ) vertices connected to every other vertex and still any square of a Hamilton cycle has absolutediscrepancy ๐ ( ๐ ) . Our proof follows a very similar structure to that of Balogh, Csaba, Pluhยดar and Treglown [2] for ๐พ ๐ -tilings.We start by applying the regularity lemma on ๐บ to obtain the reduced graph ๐ and the corresponding edgelabelling ๐ ๐ . Before proving the conjecture for large ๐, Komlยดos, Sยดarkยจozy and Szemerยดedi [11], proved an approximateversion, namely they proved it for ๐ -vertex graphs with minimum degree at least ( โ /( ๐ + ) + ๐ ) ๐. Wemake slight modi๏ฌcations to their proof to establish two important claims.We prove that a ๐พ ๐ + -tiling of ๐ with linear discrepancy with respect to ๐ ๐ can be used to construct an ๐ ๐กโ power of a Hamilton cycle in ๐บ with linear discrepancy with respect to ๐ . Combined with Theorem 1.3,this is enough to deduce the case ๐ = ๐ : suppose that ๐น is a small subgraph of ๐ and there are two ๐ถ ๐ -templates of ๐น covering the vertices the same number of times, but having di๏ฌerent discrepancies with6espect to ๐ ๐ . Then in ๐บ there exists the ๐ ๐กโ power of a Hamilton cycle with linear discrepancy with respectto ๐ . From this point on, we only โworkโ on the reduced graph ๐ . To use the last claim, we need a subgraph ๐น onwhich we can ๏ฌnd two di๏ฌerent ๐ถ ๐ -templates, so the simplest subgraph we can study is an ( ๐ + ) -clique.We prove that every ( ๐ + ) -clique in ๐ is either a copy of ๐พ + ๐ + , ๐พ โ ๐ + , ( ๐พ ๐ + , +) -star or ( ๐พ ๐ + , โ) -star. As ๐ has large minimum degree, every clique of size ๐ โค ๐ + ๐ + . Thisshows that every clique of size ๐ โค ๐ + ๐พ + ๐ , ๐พ โ ๐ , ( ๐พ ๐ , +) -star or ( ๐พ ๐ , โ) -star.We assume ๐บ has no ๐ ๐กโ power of a Hamilton cycle with large absolute discrepancy. By the Hajnal-Szemerยดedi theorem, we can ๏ฌnd a ๐พ ๐ + -tiling T of ๐ . The previous arguments show that only four typesof cliques appear in T and T has a small discrepancy with respect to ๐ ๐ . 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 di๏ฌerent types. For several relevant cases when there are many edgesbetween the two cliques, we construct two ๐ถ ๐ -templates of di๏ฌerent discrepancies, which contradicts ourassumption by the claim about ๐ถ ๐ -templates. Finally, this restricts the number of edges between di๏ฌerentcliques which leads to a contradiction with the minimum degree assumption on ๐ . ๐ถ ๐ -tilings of ๐ In [11] Komlยดos, Sยดarkยจozy and Szemerยดedi proved an approximate version of the Pยดosa-Seymour conjecture.More precisely, they show that for any ๐ > ๐บ on ๐ vertices, for su๏ฌciently large ๐ , with minimumdegree at least ( ๐ /( ๐ + )+ ๐ ) ๐ contains the ๐ ๐กโ power of a Hamilton cycle. Their argument starts by applyingthe regularity lemma to get an ( ๐, ๐ / ) -super-regular partition ( ๐ ๐ ) ๐ก๐ = of ๐ ( ๐บ ) and the Hajnal-Szemerยดeditheorem to obtain a ๐พ ๐ + -tiling of its ( ๐, ๐ / ) -reduced graph ๐ . Let ๐พ , . . . , ๐พ ๐ be the ( ๐ + ) -cliques in thistiling. Then they proceed to ๏ฌnd short paths ๐ , . . . , ๐ ๐ in ๐บ , each ๐ ๐ โconnectingโ subsequent cliques ๐พ ๐ , ๐พ ๐ + in the tiling, and โattachโ the exceptional verticesโthe ones in ๐ together with some other vertices notrespecting a certain degree conditionโto these paths. Finally, for each clique ๐พ ๐ , the Blow-up Lemma [10]is applied to ๏ฌnd the ๐ ๐กโ power of a Hamilton path on the set of unused vertices in ๐บ corresponding to thisclique, which together with paths ๐ , . . . , ๐ ๐ closes the ๐ ๐กโ power of a Hamilton cycle.We need a slightly more general result for our application. Instead of a ๐พ ๐ + -tiling in ๐ , we assume a ๐ถ ๐ -tiling T of ๐ is given in which most of the cycles ๐ถ ๐ โ T have length ๐ +
1. Then we proceed similarlyas outlined above, where for each ๐ถ ๐ โ T we choose an ( ๐ + ) -clique in ๐ถ ๐๐ to represent ๐พ ๐ for which theconnecting path ๐ ๐ is constructed. In order to argue about the discrepancy of the found ๐ ๐กโ power of aHamilton cycle ๐ป ๐ in ๐บ , we explicitly state a property of the construction in [11], which is that most ofthe edges in ๐ป ๐ come from the given tiling and that these edges are used in a balanced way. Intuitively, ifthe ๐ถ ๐ -tiling of ๐ is given together with a function ๐ ๐ as in (1), then the discrepancy of the found ๐ป ๐ in ๐บ is at most ๐ผ๐ away from ๐ ยท ๐ ๐ (T ) , for an arbitrarily small ๐ผ >
0, where we remind the reader that ๐ isthe size of each cluster. Additionally, we show that, given two similar ๐ถ ๐ -tilings, we can ๏ฌnd similar ๐ ๐กโ powers of a Hamilton cycle. We make these notions precise in the following statement. Proposition 5.1.
For any integer ๐ โฅ and any ๐ผ, ๐ > there exist โ โ N and ๐ > such that the followingholds. Suppose ๐บ is a graph on ๐ vertices with ๐ฟ ( ๐บ ) โฅ ( ๐ /( ๐ + ) + ๐ ) ๐ and let ๐ : ๐ธ ( ๐บ ) โ {โ , } be itsedge labelling. Let ( ๐ ๐ ) โ๐ = , where โ โฅ โ and ( ๐ + ) | โ, be an ( ๐, ๐ / ) -super-regular partition of ๐ ( ๐บ ) and ๐ its ( ๐, ๐ / ) -reduced graph with ๐ ๐ : ๐ธ ( ๐ ) โ {โ , } as de๏ฌned in (1) . Suppose we are given two ๐ถ ๐ -tilings T = K . โช C and T = K . โช C of ๐ such that โข K consists only of ( ๐ + ) -cycles, and |C ๐ | โค ๐ , for ๐ โ { , } .Then there exist ๐ ๐กโ powers of Hamilton cycles ๐ป ๐ , ๐ป ๐ โ ๐บ such that(i) | ๐ ( ๐ป ๐๐ ) โ ๐๐ ๐ (T ๐ ) | โค ๐ผ๐ , for ๐ โ { , } , and(ii) | ๐ ( ๐ป ๐ ) โ ๐ ( ๐ป ๐ ) | โฅ ๐ | ๐ ๐ (C ) โ ๐ ๐ (C ) | โ ๐ถ๐ ,where ๐ถ > is an absolute constant.Proof. The proof is almost a one-to-one copy of the argument by Komlยดos, Sยดarkยจozy and Szemerยดedi in [11].We thus only give a rough sketch without spelling out all the details but speci๏ฌcally 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 / ๐ โช ๐ผ โช / โ โช ๐ โช ๐ โช / ๐ . Recall that | ๐ | โค ๐๐ and | ๐ ๐ | = ๐ for 1 โค ๐ โค โ. Let K = { ๐ถ ๐ , . . . , ๐ถ ๐๐ } and C = { ๐ถ ๐๐ + , . . . , ๐ถ ๐๐ + ๐ } where ๐ โค โ๐ + and ๐ โค ๐ . The cycles ๐ถ , . . . , ๐ถ ๐ are of size ๐ + , that is, ๐ถ ๐ , . . . , ๐ถ ๐๐ are ( ๐ + ) -cliques whereeach edge has multiplicity 2 . For ๐ โ [ ๐ + ๐ ] , let ๐ ๐ , ๐ ๐ , . . . , ๐ ๐๐ + be the ๏ฌrst ๐ + ๐ถ ๐ and let ๐พ ๐ denote the clique on these ๐ + P ๐ between cliques ๐พ ๐ and ๐พ ๐ + for all ๐ โ [ ๐ + ๐ ] , where we denote ๐พ ๐ + ๐ + = ๐พ . In [11] it is shown that in ๐บ there exists the ๐ ๐กโ power of a path on vertices ๐ , ๐ , . . . , ๐ ๐ก with ๐ก = ๐ ( ๐ ) connecting cliques ๐พ ๐ and ๐พ ๐ + such that ๐ ๐ โ ๐ ๐๐ and ๐ ๐ก โ ๐ โ + ๐ โ ๐ ๐ + ๐ for ๐ โ [ ๐ + ] . Additionally, these vertices can be chosen such that | ๐ ( ๐ , ๐ , . . . , ๐ ๐ ) โฉ ๐ ๐๐ + | > ( ๐ โ ๐ ) ๐ ๐ and | ๐ ( ๐ ๐ก , ๐ ๐ก โ , . . . , ๐ ๐ก + โ ๐ ) โฉ ๐ ๐ + ๐ + โ ๐ | > ( ๐ โ ๐ ) ๐ ๐, โ ๐ โ [ ๐ ] , (2)which ensures we can later extend this to an ๐ ๐กโ power of a Hamilton cycle in ๐บ .
An important thing tonote is that these paths can be constructed one by one such that the paths P , P , . . . , P ๐ โ only depend on K . The arguments from [11] translate directly to our case.Next we add some more vertices to the exceptional set ๐ . From a cluster ๐ ๐๐ in a cycle ๐ถ ๐ we move to ๐ all vertices ๐ฃ not used on the paths P , . . . , P ๐ โ for which there is a ๐ โฒ such that { ๐ ๐๐ , ๐ ๐๐ โฒ } โ ๐ธ ( ๐ถ ๐ ) and deg ( ๐ฃ, ๐ ๐๐ โฒ ) โค ๐ | ๐ ๐๐ โฒ | , where we consider the original clusters ๐ ๐๐ โฒ , that is, we also consider the vertices already used in theconnecting paths in this calculation. Because of ( ๐, ๐ ) -regularity and because | ๐ถ ๐ | โค ๐ for all ๐ โ [ ๐ + ๐ ] by de๏ฌnition of a ๐ถ ๐ -tiling, there are at most | ๐ถ ๐ | ยท ๐ | ๐ ๐๐ | โค ๐ ๐๐ such vertices in each cluster ๐ ๐๐ . Then wemove the smallest possible number of verties to ๐ so that each cluster has the same number of vertices.We denote the new number of vertices per cluster by ๐ โฒ and we still write ๐ for the enlarged exceptionalset which now satis๏ฌes | ๐ | โค ๐ ๐๐. For each vertex ๐ฃ โ ๐ we ๏ฌnd all cliques ๐พ ๐ with ๐ โ [ ๐ โ ] suchthat deg ( ๐ฃ, ๐ ๐๐ ) โฅ ๐ | ๐ ๐๐ | , โ ๐ โ [ ๐ + ] . Again, in the above we consider the original clusters ๐ ๐๐ . Let ๐ฅ denote the number of such clusters. Wehave (cid:18) โ ๐ + + ๐ (cid:19) ๐ โค deg ( ๐ฃ ) โค ๐ + ๐ ร ๐ | ๐ถ ๐ | ! ๐ โฒ + | ๐ | + ๐ฅ ( ๐ + ) ๐ โฒ + ( ๐ โ โ ๐ฅ ) (cid:16) ๐๐ โฒ + ๐ ๐ โฒ (cid:17) . Using ๐ โค ๐ , | ๐ถ ๐ | โค ๐ and that โ is su๏ฌciently large, a simple calculation yields ๐ฅ โฅ ๐โ / . We assign ๐ฃ to one of these cliques such that no clique is assigned too many vertices. A greedy algorithm assignsat most ๐ ๐ โฒ vertices to each clique for ๐ = ๐ ๐๐ . Now, we extend the path P ๐ to contain all verticesassigned to clique ๐พ ๐ by using additional three vertices from each of the clusters ๐ ๐๐ , ๐ โ [ ๐ + ] per addedvertex. This can be done without changing the ๏ฌrst and last ๐ vertices on the path so that property 2 is stillsatis๏ฌed. 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 ( โ ๐ ) ๐ โฒ โฅ ( โ ๐ผ ๐ ) ๐. (3)We are in the same situation as in [11]: for each cycle ๐ถ ๐ we have the same number of remaining vertices ineach ๐ ๐๐ and the endpoints of the connecting paths satisfy 2, so we can apply Lemma 2.8 and the subsequentremark to close the ๐ ๐กโ power of a Hamilton cycle ๐ป ๐ . From 3, it easily follows that ๐ป ๐ satis๏ฌes (i). Indeed,for every edge ( ๐ ๐ฅ , ๐ ๐ฆ ) in some ๐ถ ๐๐ , ๐ โ [ ๐ + ๐ ] , we included in ๐ป ๐ at least mul ๐ถ ๐ ( ๐ ๐ฅ , ๐ ๐ฆ ) ยท ( โ ๐ผ ๐ ) ๐ corresponding edges of color ๐ ๐ ( ๐ ๐ฅ , ๐ ๐ฆ ) . Apart from these, there are at most (| ๐ | + ๐ผ ๐ ๐ ) ยท ๐ โค ๐ผ ๐ edgesin ๐ป ๐ . Hence, | ๐ ( ๐ป ๐ ) โ ๐๐ ๐ (T ) | โค ๐ผ ๐ ๐ | ๐ ๐ (T ) | + ๐ผ ๐ โค ๐ผ ๐ ๐ ยท ๐ โ + ๐ผ ๐ โค ๐ผ๐. Now suppose we are given two tilings T , T as in the statement of the proposition. We construct two ๐ ๐กโ powers of a Hamilton cycle ๐ป ๐ and ๐ป ๐ with a small modi๏ฌcation to the above procedure to ensure theydo not di๏ฌer too much. We think of the above algorithm as three stages of adding edges to a subgraphwhich eventually becomes the ๐ ๐กโ power of a Hamilton cycle. We use I and I to denote the runs of thealgorithm on T = T and T = T , respectively.The ๏ฌrst stage of connecting the cliques is done exactly as above. Note that the paths P , . . . , P ๐ โ are thesame in both instances. Let ๐ denote the exceptional set and ๐ the set of vertices used for the connectingpaths in I . Analogously de๏ฌne ๐ and ๐ with respect to I . Note that | ๐ ฮ ๐ | โค ( ๐ + ) ยท ๐ ( ๐ ) = ๐ ( ๐ ) because the two runs only deviate after ๏ฌnding the ๏ฌrst ๐ โ ๐ ( ๐ ) . Sincethe degree of any vertex in the ๐ ๐กโ power of a Hamilton cycle is 2 ๐, this implies the two subgraphs di๏ฌerin ๐ ( ๐ ) edges at this point.We can assume that the new number of vertices per cluster ๐ โฒ is the same in I and I , otherwise simplyadd a few vertices to ๐ ๐ if ๐ โฒ is larger in I ๐ . Set ๐ = ๐ โช ๐ and note that now | ๐ | โค ๐ ๐๐ which doesnot a๏ฌect the above argument. To make the two found subgraphs similar, we treat ๐ \ ๐ and ๐ \ ๐ asthe new exceptional sets for I and I , respectively. In the second stage, we add the exceptional verticesto paths P , . . . , P ๐ โ . 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 ๐ \ ( ๐ โช ๐ ) to the same cliques for the tworuns. We can then embed the vertices from ๐ \ ( ๐ โช ๐ ) in exactly the same way in I and I . Thus I and I only deviate after embedding all but at most | ๐ ฮ ๐ | = ๐ ( ๐ ) vertices. Each added vertex extendsthe path by ๐ ( ๐ ) vertices, so this stage introduces at most ๐ ( ๐ ) edges to ๐ป ๐ ฮ ๐ป ๐ . Finally, in the third stage, we ๏ฌnd connecting paths inside cliques ๐พ ๐ . The edges in these paths have thesame labels in I and I for all cliques ๐พ ๐ , ๐ โ [ ๐ ] . The number of vertices left in cliques ๐พ , . . . , ๐พ ๐ mightdi๏ฌer in I and I at this point, but in total only by | ๐ ฮ ๐ | ยท ๐ ( ๐ ) . Hence, the di๏ฌerence between thevalues of edges in cliques ๐พ , . . . , ๐พ ๐ , introduced at this stage is at most ๐ ( ๐ ) . Since for each edge in C weadd ๐ โฒ โฅ ( โ ๐ผ ๐ ) ๐ edges to ๐ป ๐ and analogously for C and ๐ป ๐ , we have | ๐ ( ๐ป ๐ ) โ ๐ ( ๐ป ๐ ) | โฅ (cid:16) โ ๐ผ ๐ (cid:17) ๐ ยท | ๐ ๐ (C ) โ ๐ ๐ (C ) | โ ๐ ( ๐ ) โฅ ๐ (cid:12)(cid:12) ๐ ๐ (C ) โ ๐ ๐ (C ) (cid:12)(cid:12) โ ๐ถ๐ . (cid:3) It is enough to prove the theorems for ๐ โช / ๐ . We de๏ฌne additional constants ๐พ, ๐, ๐ฝ, ๐ > ๐ , โ , ๐ฟ โ N such that 0 < / ๐ โช ๐พ โช / ๐ฟ โค / โ โช ๐ โช ๐ฝ โช ๐ = ๐ / โช / ๐, ๐ฟ is the constant obtained from Lemma 2.5 with parameters ๐, โ . Let ๐บ be a graph with ๐ โฅ ๐ vertices and ๐ : ๐ธ ( ๐บ ) โ {โ , } an edge labelling as in the statement of the theorem.Both proofs start by applying Lemma 2.5 to ๐บ with parameters ๐, ๐ and โ . We thus obtain the reducedgraph ๐ whose vertices are clusters ๐ , ๐ , . . . , ๐ โ , where each of the clusters ๐ ๐ is of size ๐. We also havethe exceptional set ๐ of size at most ๐๐. Recall that ๐ + โ. The reduced graph ๐ inherits the edgelabelling ๐ ๐ 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 ๐พ ๐ + -tiling T of ๐ such that | ๐ ๐ (T ) | โฅ ๐ฝโ. Then, in ๐บ there exists the ๐ ๐กโ power of a Hamilton cycle ๐ป ๐ satisfying | ๐ ( ๐ป ๐ ) | โฅ ๐พ๐. (cid:3) Claim 6.2.
Let ๐น be a subgraph of ๐ on at most ๐ vertices and let F = { ๐ถ , ๐ถ , . . . , ๐ถ ๐ } and F = { ๐ถ , ๐ถ , . . . , ๐ถ ๐ } be two ๐ถ ๐ -templates of ๐น such that each vertex of ๐น appears exactly ๐ times in F and ๐ times in F for some ๐ โค ๐ . If F and F have di๏ฌerent discrepancies with respect to ๐ ๐ , then in ๐บ thereexists the ๐ ๐กโ power of a Hamilton cycle ๐ป ๐ satisfying | ๐ ( ๐ป ๐ ) | โฅ ๐พ๐. Proof.
Split each of the clusters ๐ ๐ of the regular partition into ๐ clusters of size ๐ โฒ = โ ๐ / ๐ โ and put theremaining vertices in ๐ . Let ๐ ( ๐ ๐ ) denote the set of ๐ clusters formed by ๐ ๐ . De๏ฌne ๐ โฒ as a blow-up of ๐ with the edge coloring ๐ ๐ โฒ in the natural way: if ๐ ๐ and ๐ ๐ were joined by an edge, then put a completebipartite graph between ๐ ( ๐ ๐ ) and ๐ ( ๐ ๐ ) with all edges of color ๐ ๐ ( ๐ ๐ , ๐ ๐ ) . Let ๐น โฒ denote the correspondingblow-up of ๐น .
Formally, we de๏ฌne ๐ ( ๐น โฒ ) = ร ๐ ๐ โ ๐ ( ๐น ) ๐ ( ๐ ๐ ) and ๐ธ ( ๐น โฒ ) = ร ( ๐ ๐ ,๐ ๐ ) โ ๐ธ ( ๐น ) ๐ โฒ [ ๐ ( ๐ ๐ ) , ๐ ( ๐ ๐ )] . ByLemma 2.7, we get that all clusters joined by an edge in ๐ โฒ form an ๐ โฒ regular pair in ๐บ for ๐ โฒ = ๐๐ andhave density at least ๐ โฒ = ๐ โ ๐. For the new exceptional set ๐ we have | ๐ | โค ๐ โฒ ๐. Note that ๐ โฒ satis๏ฌes ๐ฟ ( ๐ โฒ ) โฅ ( โ ๐ + + ๐ ) | ๐ โฒ | and so ๐ฟ ( ๐ โฒ \ ๐น โฒ ) โฅ ( โ ๐ + ) | ๐ โฒ | . Applying Theorem 2.4, weobtain a ๐พ ๐ + -tiling K of ๐ โฒ \ ๐น โฒ . Using F we construct a ๐ถ ๐ -tiling C of ๐น โฒ as follows. In the cycles in F , simply replace every occurrence of a vertex ๐ ๐ with a di๏ฌerent vertex from ๐ ( ๐ ๐ ) . It is easy to verify that C is a ๐ถ ๐ -tiling of ๐น โฒ . Analogously, we construct C from F . Let T = K . โช C and T = K . โช C . We apply Proposition 5.1 to obtain two ๐ ๐กโ powers of Hamilton cycles ๐ป ๐ and ๐ป ๐ which satisfy (ii) with respect to ๐ โฒ and ๐ ๐ โฒ . Finally, note that ๐ ๐ โฒ (T ๐ ) = ๐ ๐ โฒ (K) + ๐ ๐ โฒ (C ๐ ) and ๐ ๐ โฒ (C ๐ ) = ๐ ๐ (F ๐ ) for ๐ = , . Hence, we have | ๐ ( ๐ป ๐ ) โ ๐ ( ๐ป ๐ ) | โฅ ๐ โฒ | ๐ ๐ โฒ (T ) โ ๐ ๐ โฒ (T ) | โฅ ๐ ( ๐ + ) ยท โฅ ๐พ๐. Therefore, at least one of ๐ป ๐ , ๐ป ๐ has absolute discrepancy at least ๐พ๐. (cid:3) First we resolve the case ๐ = Proof of Theorem 1.5.
Recall that ๐ฟ ( ๐บ ) โฅ ( / + ๐ ) ๐, so by Fact 2.6, we get ๐ฟ ( ๐ ) โฅ ( / + ๐ / ) | ๐ | . Let ๐ฝ bethe value of ๐พ given by Theorem 1.3 with parameters ๐ = ๐ / . Applying Theorem 1.3 to the reducedgraph ๐ we obtain a ๐พ -tiling T of ๐ of absolute discrepancy at least ๐ฝโ. By Claim 6.1, ๐บ contains thesquare of a Hamilton cycle with absolute discrepancy at least ๐พ๐. (cid:3) In the rest of the paper, we ๏ฌnish the proof of Theorem 1.4.
Proof of Theorem 1.4.
We prove the theorem by contradiction, so we assume that ๐บ does not contain the ๐ ๐กโ power of a Hamilton cycle with absolute discrepancy at least ๐พ๐. Recall that we have ๐ โฅ ๐ฟ ( ๐บ ) โฅ (cid:0) โ ๐ + + ๐ (cid:1) ๐. By Fact 2.6, the reduced graph satis๏ฌes ๐ฟ ( ๐ ) โฅ (cid:18) โ ๐ + + ๐ (cid:19) โ. (4)10is yields the following simple observation. Claim 6.3.
For any ๐ฃ , ๐ฃ , . . . , ๐ฃ ๐ + โ ๐ , we have | ๐ ( ๐ฃ , ๐ฃ , . . . , ๐ฃ ๐ + ) | โฅ ( ๐ + ) ๐โ / . (cid:3) The following claim shows that ๐ is highly structured with respect to ๐ ๐ . Claim 6.4.
Let ๐พ be a clique in ๐ of size ๐ โค ๐ + . Then ๐พ is a copy of one of the following: ๐พ + ๐ , ๐พ โ ๐ , ( ๐พ ๐ , +) -staror ( ๐พ ๐ , โ) -star.Proof. First we prove the claim for ( ๐ + ) -cliques. Let ๐พ = { ๐ฃ , ๐ฃ , . . . , ๐ฃ ๐ + } be an ( ๐ + ) -clique in ๐ . De๏ฌne ๐ถ = ( ๐ฃ , ๐ฃ , ๐ฃ , ๐ฃ , ๐ฃ , . . . , ๐ฃ ๐ + ) and ๐ถ = ( ๐ฃ , ๐ฃ , ๐ฃ , ๐ฃ , ๐ฃ , . . . , ๐ฃ ๐ + ) . We can view { ๐ถ } and { ๐ถ } as ๐ถ ๐ -templates on ๐พ .
Thus, by Claim 6.2, we get ๐ ๐ ( ๐ถ ๐ ) = ๐ ๐ ( ๐ถ ๐ ) . Note that ๐ ๐ ( ๐ถ ๐ ) = ร โค ๐ < ๐ โค ๐ + ๐ ๐ ( ๐ฃ ๐ , ๐ฃ ๐ ) โ ๐ + ร ๐ = ๐ ๐ ( ๐ฃ ๐ , ๐ฃ ๐ + ) , where we denote ๐ฃ ๐ + = ๐ฃ . Hence, 0 = ๐ ๐ ( ๐ถ ๐ ) โ ๐ ๐ ( ๐ถ ๐ ) = ๐ ๐ ( ๐ฃ , ๐ฃ ) + ๐ ๐ ( ๐ฃ , ๐ฃ ) โ ๐ ๐ ( ๐ฃ , ๐ฃ ) โ ๐ ๐ ( ๐ฃ , ๐ฃ ) . Asthe enumeration of the vertices was arbitrary, for any distinct ๐, ๐, ๐, ๐ โ ๐พ, the following holds: ๐ ๐ ( ๐, ๐ ) + ๐ ๐ ( ๐, ๐ ) = ๐ ๐ ( ๐, ๐ ) + ๐ ๐ ( ๐, ๐ ) . (5)The rest of the proof appears in [2]. We present a slightly shorter argument. Assume that ๐พ is notmonochromatic, so there exists a vertex ๐ฃ with ๐ + ๐พ ( ๐ฃ ) , ๐ โ ๐พ ( ๐ฃ ) โ โ . Without loss of generality, assume | ๐ + ๐พ ( ๐ฃ ) | โฅ ๐ข โ ๐ โ ๐พ ( ๐ฃ ) . Consider arbitrary ๐ฅ, ๐ฆ โ ๐ + ๐พ ( ๐ฃ ) . By (5), we get ๐ ๐ ( ๐ฅ, ๐ฃ ) + ๐ ๐ ( ๐ข, ๐ฆ ) = ๐ ๐ ( ๐ฅ, ๐ฆ )+ ๐ ๐ ( ๐ข, ๐ฃ ) . By, de๏ฌnition ๐ ๐ ( ๐ฅ, ๐ฃ ) = ๐ ๐ ( ๐ข, ๐ฃ ) = โ , so this implies ๐ ๐ ( ๐ข, ๐ฆ ) = โ ๐ ๐ ( ๐ฅ, ๐ฆ ) = . If | ๐ โ ๐พ ( ๐ฃ ) | โฅ , a completely analogous argument shows ๐ ๐ ( ๐ข, ๐ฆ ) = , a contradiction. From this weconclude ๐ โ ๐ ( ๐ฃ ) = { ๐ข } . Applying the same reasoning to every pair ๐ฅ, ๐ฆ โ ๐ + ๐ ( ๐ฃ ) , we get ๐ ( ๐ฅ, ๐ฆ ) = ๐ ( ๐ข, ๐ฆ ) = โ ๐ฅ, ๐ฆ โ ๐ + ๐ ( ๐ฃ ) . In other words, ๐พ is a ( ๐พ ๐ + , โ) -star with ๐ข as its head.Now, suppose ๐พ is a clique in ๐ of size ๐ โค ๐ + . By Claim 6.3, ๐พ can be extended, vertex by vertex, tosome clique ๐พ โฒ of size ๐ + . The statement now easily follows from the result for ( ๐ + ) -cliques. (cid:3) By (4), ๐ has large minimum degree, so we can apply Theorem 2.4 to obtain a ๐พ ๐ + -tiling T of ๐ . FromClaim 6.4 we conclude there are only four types of cliques in T . Let ๐ด denote the set of ๐พ + ๐ + in T ; ๐ต theset of ๐พ โ ๐ + in T ; ๐ถ the set of ( ๐พ ๐ + , +) -stars in T ; and ๐ท the set of ( ๐พ ๐ + , โ) -stars in T . Without loss ofgenerality, we may assume | ๐ต | + | ๐ถ | โฅ | ๐ด | + | ๐ท | . (6)Under the assumption that ๐บ does not have ๐ ๐กโ powers of Hamilton cycles with large discrepancy, weestablish several claims about edges between cliques of di๏ฌerent types. We state these claims here anddefer their proofs to the end of the paper. Claim 6.5.
Consider a vertex ๐ฅ of a clique ๐ โ ๐ด and let ๐ be a copy of ๐พ โ ๐ + in ๐ต. Then we may assume deg ( ๐ฅ , ๐ ) โค ๐ โ . Claim 6.6.
Consider a vertex ๐ฅ of a clique ๐ โ ๐ด and let ๐ be a ( ๐พ ๐ + , +) -star in ๐ถ. Then we may assume deg ( ๐ฅ , ๐ ) โค ๐ โ . Claim 6.7.
Suppose ๐ฅ is the head of a clique ๐ โ ๐ถ and let ๐ โ ๐ท .
Then, we may assume deg ( ๐ฅ , ๐ ) โค ๐ โ . ๐บ has no ๐ ๐กโ power of a Hamilton cycle with absolute discrepancy at least ๐พ๐. We can assume | ๐ ๐ (T ) | < ๐ฝโ, asotherwise we can ๏ฌnd the desired ๐ ๐กโ power of a Hamilton cycle by Claim 6.1.First, we show that ๐ด = โ . Otherwise, consider some vertex ๐ฃ of a clique in ๐ด. By Claims 6.5 and 6.6, ๐ฃ canhave at most ๐ โ ๐ต โช ๐ถ. Trivially, it can have at most ๐ + ๐ด โช ๐ท .
Clearly, | ๐ด | + | ๐ต | + | ๐ถ | + | ๐ท | = โ /( ๐ + ) . Using (6), we getdeg ( ๐ฃ ) โค ( ๐ โ ) (| ๐ต | + | ๐ถ |) + ( ๐ + ) (| ๐ด | + | ๐ท |) โค ๐๐ + โ, which contradicts the degree assumption (4). Hence, ๐ด = โ . Note that | ๐ต | + | ๐ถ | โ | ๐ท | โค ๐ฝโ, as otherwise | ๐ ๐ (T ) | โฅ ๐ฝโ. Hence, we have 2 (| ๐ต | + | ๐ถ |) โค (cid:0) ๐ + + ๐ฝ (cid:1) โ. Suppose ๐ถ is nonempty and let ๐ฃ be the head of some ( ๐พ ๐ + , +) -star in ๐ถ. By Claim 6.7, it can have at most ๐ โ ๐ท .
Therefore,deg ( ๐ฃ ) โค ( ๐ โ ) | ๐ท | + ( ๐ + ) (| ๐ต | + | ๐ถ |) = ( ๐ โ ) โ๐ + + (| ๐ต | + | ๐ถ |) โค (cid:18) โ ๐ + + ๐ฝ (cid:19) โ. This contradicts (4), so ๐ถ = โ . Finally, from (6), we have | ๐ต | โฅ | ๐ท | which yields ๐ ๐ (T ) = โ (cid:18) ๐ + (cid:19) | ๐ต | + (cid:18) (cid:18) ๐ + (cid:19) โ ๐ (cid:19) | ๐ท | โค โ ๐ฝโ, a contradiction. (cid:3) Proofs of Claims 6.5, 6.6 and 6.7
In the following we will consider two ( ๐ + ) -cliques in T which we denote by ๐ = { ๐ฅ , . . . , ๐ฅ ๐ + } and ๐ = { ๐ฆ , . . . , ๐ฆ ๐ + } . Additionally, we assume that ๐ฅ has at least ๐ edges towards vertices of ๐ .
Without lossof generality, we assume ๐ฅ is connected to ๐ฆ , ๐ฆ , . . . , ๐ฆ ๐ . From Claim 6.3 it follows that there is a vertex ๐ฅ โฒ such that ๐ โฒ = ๐ โช { ๐ฅ โฒ } forms an ( ๐ + ) -clique. Similarly, there is a vertex ๐ฆ โฒ such that ๐ โฒ = ๐ โช { ๐ฆ โฒ } formsan ( ๐ + ) -clique and all the vertices ๐ฅ , . . . ๐ฅ ๐ + , ๐ฅ โฒ , ๐ฆ , . . . , ๐ฆ ๐ + , ๐ฆ โฒ are distinct. We construct two templateson ๐น = ๐ โฒ . โช ๐ โฒ and show, for the cases mentioned in the claims, that these templates have di๏ฌerentdiscrepancy. By Claim 6.2, this contradicts the assumption that ๐บ has no ๐ ๐กโ power of a Hamilton cyclewith a large discrepancy. We start by de๏ฌning four cycles which will be used in the templates. ๐ถ = ( ๐ฅ , ๐ฅ , . . . , ๐ฅ ๐ + , ๐ฅ โฒ ) ๐ถ = ( ๐ฅ , ๐ฅ , ๐ฅ , . . . , ๐ฅ ๐ + , ๐ฅ โฒ ) ๐ถ = ( ๐ฆ , ๐ฆ , ๐ฆ , . . . , ๐ฆ ๐ + , ๐ฆ โฒ ) ๐ถ = ( ๐ฅ , ๐ฆ , ๐ฆ , . . . , ๐ฆ ๐ ,๐ฆ ๐ + , ๐ฆ โฒ , ๐ฆ , ๐ฆ , . . . , ๐ฆ ๐ โ , ๐ฆ ๐ โ ,๐ฆ ๐ + , ๐ฆ โฒ , ๐ฆ , ๐ฆ , . . . , ๐ฆ ๐ โ , ๐ฆ ๐ ,. . . ,๐ฆ ๐ + , ๐ฆ โฒ , ๐ฆ , ๐ฆ , . . . , ๐ฆ ๐ ,๐ฆ ๐ + , ๐ฆ โฒ , ๐ฆ , ๐ฆ , . . . , ๐ฆ ๐ ,๐ฆ ๐ + , ๐ฆ โฒ , ๐ฆ , ๐ฆ , . . . , ๐ฆ ๐ ) Using these, we de๏ฌne two templates F and F as follows: F = (cid:0) ( ๐ + ) ร ๐ถ , ( ๐ + ) ร ๐ถ (cid:1) and F = (cid:0) ๐ถ , ๐ ร ๐ถ , ๐ถ (cid:1) , ๐ ร ๐ถ ๐ to indicate ๐ copies of ๐ถ ๐ . Note that F contains each vertex in ๐น exactly ๐ + ๐ โฒ \ { ๐ฅ } appears once in ๐ถ and ๐ถ , so F contains each of these vertices ๐ + ๐ถ contains ๐ฅ once and each of the vertices in ๐ โฒ exactly ๐ + | ๐ถ | = + ( ๐ + ) ( ๐ + ) = ๐ + ๐ + , (7)which will be useful later on. Additionally, it is easy to see that F contains each vertex in ๐น exactly ๐ + F and F have di๏ฌerent 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 ๐ ๐กโ power of a cycle, we will mostly use the following principle. As the ( ๐ + ) -cliques weconsider are highly structured, most of the edge values under consideration are known given the types ofcliques of ๐ and ๐ .
More precisely, we ๏ฌnd a small subset of edges ๐ธ โฒ such that all edges in ๐ธ ( ๐ถ ๐๐ ) \ ๐ธ โฒ havethe same color ๐ . Then, we can calculate the discrepancy of ๐ถ ๐๐ as ๐ ๐ ( ๐ถ ๐๐ ) = ๐ (cid:0) ๐ | ๐ถ ๐ | โ | ๐ธ โฒ | (cid:1) + ๐ ๐ ( ๐ธ โฒ ) . Additionally, observe that ๐ ๐ (F ) โ ๐ ๐ (F ) = โ ๐ ๐ ( ๐ถ ๐ ) + ๐ ๐ ( ๐ถ ๐ ) + ( ๐ + ) ๐ ๐ ( ๐ถ ๐ ) โ ๐ ๐ ( ๐ถ ๐ ) . (8)Finally, we proceed to prove the individual claims. Claim 6.5.
Consider a vertex ๐ฅ of a clique ๐ โ ๐ด and let ๐ be a copy of ๐พ โ ๐ + in ๐ต. Then we may assume deg ( ๐ฅ , ๐ ) โค ๐ โ . Proof.
Suppose deg ( ๐ฅ , ๐ ) โฅ ๐ and let F , F be de๏ฌned as above. Applying Claim 6.4 to the clique ๐ โฒ , weget that all edges from ๐ฅ โฒ to ๐ have the same color ๐ ๐ ( ๐ฅ โฒ , ๐ฅ ) . Analogously, all edges from ๐ฆ โฒ to ๐ havecolor ๐ ๐ ( ๐ฆ โฒ , ๐ฆ ) and all edges from ๐ฅ to ๐ \ { ๐ฆ ๐ + } have color ๐ ๐ ( ๐ฅ , ๐ฆ ) . We calculate: ๐ ๐ ( ๐ถ ๐ ) = ( ๐ + ) ๐ โ ๐ + ๐ ๐ ๐ ( ๐ฅ โฒ , ๐ฅ ) = ๐ โ ๐ + ๐ ๐ ๐ ( ๐ฅ โฒ , ๐ฅ ) ๐ ๐ ( ๐ถ ๐ ) = ( ๐ + ) ๐ โ ๐ + ๐ ๐ ๐ ( ๐ฅ โฒ , ๐ฅ ) = ๐ + ๐ ๐ ๐ ( ๐ฅ โฒ , ๐ฅ ) ๐ ๐ ( ๐ถ ๐ ) = โ( ( ๐ + ) ๐ โ ๐ ) + ๐ ๐ ๐ ( ๐ฆ โฒ , ๐ฆ ) = โ ๐ + ๐ ๐ ๐ ( ๐ฆ โฒ , ๐ฆ ) ๐ ๐ ( ๐ถ ๐ ) = โ (cid:0) ๐ | ๐ถ | โ ๐ โ ๐ ( ๐ + ) (cid:1) + ๐ ๐ ๐ ( ๐ฅ , ๐ฆ ) + ๐ ( ๐ + ) ๐ ๐ ( ๐ฆ โฒ , ๐ฆ ) = โ ๐ โ ๐ + ๐ + ๐ ๐ ๐ ( ๐ฅ , ๐ฆ ) + ๐ ( ๐ + ) ๐ ๐ ( ๐ฆ โฒ , ๐ฆ ) . Plugging these values into (8), we obtain: ๐ ๐ (F ) โ ๐ ๐ (F ) = โ ๐ ๐ ๐ ( ๐ฅ , ๐ฆ ) โ . We are done by Claim 6.2. (cid:3)
Claim 6.6.
Consider a vertex ๐ฅ of a clique ๐ โ ๐ด and let ๐ be a ( ๐พ ๐ + , +) -star in ๐ถ. Then we may assume deg ( ๐ฅ , ๐ ) โค ๐ โ . Proof.
Assume deg ( ๐ฅ , ๐ ) โฅ ๐ and de๏ฌne F , F as above. By Claim 6.4, all edges from ๐ฅ โฒ to ๐ have thesame color ๐ ๐ ( ๐ฅ โฒ , ๐ฅ ) . By Claim 6.4, ๐ โฒ is a ( ๐พ ๐ + , +) -star with its head in ๐ .
Note that the values of ๐ ๐ ( ๐ถ ๐ ) and ๐ ๐ ( ๐ถ ๐ ) are as in the previous claim; we also calculate ๐ ๐ ( ๐ถ ๐ ) : ๐ ๐ ( ๐ถ ๐ ) = ๐ โ ๐ + ๐ ๐ ๐ ( ๐ฅ โฒ , ๐ฅ ) ๐ ๐ ( ๐ถ ๐ ) = ๐ + ๐ ๐ ๐ ( ๐ฅ โฒ , ๐ฅ ) ๐ ๐ ( ๐ถ ๐ ) = โ (cid:0) ( ๐ + ) ๐ โ ๐ (cid:1) + ๐ = โ ๐ ( ๐ โ ) . Now, we consider two cases:(a) ๐ฆ ๐ + is the head of ๐ .
Applying Claim 6.4 to ( ๐ \ { ๐ฆ ๐ + }) โช { ๐ฅ } , we conclude that all edges from ๐ฅ to ๐ฆ ๐ , ๐ โ [ ๐ ] are ofcolor ๐ ๐ ( ๐ฅ , ๐ฆ ) . Using this, we obtain: ๐ ๐ ( ๐ถ ๐ ) = โ( ๐ | ๐ถ | โ ๐ โ ๐ ( ๐ + )) + ๐ ๐ ๐ ( ๐ฅ , ๐ฆ ) + ๐ ( ๐ + ) โ ๐ + ๐ + ๐ + ๐ ๐ ๐ ( ๐ฅ , ๐ฆ ) . Substituting into (8), we have ๐ ๐ (F ) โ ๐ ๐ (F ) = โ ๐ ๐ ๐ ( ๐ฅ , ๐ฆ ) โ . (b) ๐ฆ ๐ + is not the head of ๐ .
Applying Claim 6.4 to ( ๐ \ { ๐ฆ ๐ + }) โช { ๐ฅ } , we get that all edges in ๐ถ ๐ incident to the head of ๐ havevalue + , while all other edges have value โ . From this we have: ๐ ๐ ( ๐ถ ๐ ) = โ( ๐ | ๐ถ | โ ๐ ( ๐ + )) + ๐ ( ๐ + ) = โ ๐ + ๐ + ๐ and ๐ ๐ (F ) โ ๐ ๐ (F ) = ๐ โ (cid:3) Claim 6.7.
Suppose ๐ฅ is the head of a clique ๐ โ ๐ถ and let ๐ โ ๐ท .
Then, we may assume deg ( ๐ฅ , ๐ ) โค ๐ โ . Proof.
Again, suppose deg ( ๐ฅ , ๐ ) โฅ ๐ and de๏ฌne F and F as above. By Claim 6.4, ๐ โฒ is a ( ๐พ ๐ + , โ) -starwith its head in ๐ .
We have ๐ ๐ ( ๐ถ ๐ ) = โ ๐ ( ๐ + ) , ๐ ๐ ( ๐ถ ๐ ) = โ ๐ ( ๐ โ ) and ๐ ๐ ( ๐ถ ๐ ) = ๐ ( ๐ โ ) . Note that the same edges of ๐ถ ๐ are known as those in Claim 6.6, but are of opposite value. Again, weconsider two cases:(a) ๐ฆ ๐ + is the head of ๐ .
Applying Claim 6.4 to ( ๐ \ { ๐ฆ ๐ + }) โช { ๐ฅ } , we conclude that all edges from ๐ฅ to ๐ฆ ๐ , ๐ โ [ ๐ ] are ofcolor ๐ ๐ ( ๐ฅ , ๐ฆ ) . So, we get ๐ ๐ ( ๐ถ ๐ ) = ๐ โ ๐ โ ๐ + ๐ ๐ ๐ ( ๐ฅ , ๐ฆ ) . Substituting into (8), we have ๐ ๐ (F ) โ ๐ ๐ (F ) = ๐ โ ๐ ๐ ๐ ( ๐ฅ , ๐ฆ ) โ . (b) ๐ฆ ๐ + is not the head of ๐ .
We apply Claim 6.4 to ( ๐ \ { ๐ฆ ๐ + }) โช { ๐ฅ } and obtain that all edges in ๐ถ ๐ incident to the head of ๐ have value โ , while all other edges have value + . From this we have: ๐ ๐ ( ๐ถ ๐ ) = ๐ โ ๐ โ ๐ and ๐ ๐ (F ) โ ๐ ๐ (F ) = ๐ โ (cid:3) Acknowledgments
The author would like to thank Miloหs Trujiยดc for introducing him to the problem, helpful discussions andmany useful comments which helped improve the presentation of this paper.
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