Proof of the Pósa−Seymour Conjecture

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In 1974 Paul Seymour conjectured that any graph G of order n and minimum degree at least (k−1)/k · n contains the (k − 1)th power of a Hamiltonian cycle. This conjecture was proved with the help of …

In 1974 Paul Seymour conjectured that any graph G of order n and minimum degree at least (k−1)/k · n contains the (k − 1)th power of a Hamiltonian cycle. This conjecture was proved with the help of the Regularity Lemma – Blow-up Lemma method for n ≥ n0 where n0 is very large. Here we present another proof that avoids the use of the Regularity Lemma and thus the resulting n0 is much smaller. The main ingredient is a new kind of connecting lemma.

Joint work with Asif Jamshed.

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  • 1. 2. Outline of the Proof Our proof has two cases. One of them is the almost extremal case, the other one is the far-from-extremal case. More precisely, we shall fix a constant a > 0 and the two cases will be: a-Extremal Condition (a-EC): A graph G is a-extremal, if there exists an A c V(G) for which ~ • ct - a)n ~ IAI ~ (i +a)n and • d(A) <a. In this case we say that the set A and the graph G are a-extremal, otherwise we say that they are a-non-extremal. Non-Extremal Case: The basic fact that we make usc of in the non- extremal case is that if we go around a complete k-partitc graph picking vertices from each of the color classes cyclically, as one can see in Fig 1, then we end up with a (k - 1)-path. We use this fact repeatedly throughout the paper. In Section 4, using the tools developed in Section 3 , we first cover a constant fraction of the vertices in G by Kk+I(t)'s and then we cover the maximum number of the remaining vertices with Kk(t)'s where t = clogn forra constant 0 < c < 1. We refer to the sets of Kk+1(t)'s a.nd Kk(t)'s by C and K respectively. We would inevitably be left with a set I of vertices that cannot be covered in such a manner. However, we show that the number ofsuch vertices is smalL Then, in Section 3.2, we prove a connecting lemma and use it to "connect" the complete (k + 1)-partite graphs and the complete k-partite graphs by (k - 1)-paths of length at most 9(k + 1)!. As noted above, after connecting the graphs inC and K we get (k - 1)th power of a cycle covering most of the vertices in V(C) U V(K) (see Figure 1). While connecting the blocks,(small complete multipartite blocks), because ~the connecting (k- 1) path has at most 9(k +1)! vertices, and - though it may use vertices from inside the blocks, that will influence at most 9(k +1)! columns. The blocks may become unbalanced and we cut them to the right size. In the process, we move a small number of vertices from V(C) U V(K) to I. To accommodate the vertices in I, using the procedure in Section 4.1.2, we insert them in b.et,ween vertices of V(C) UV(K) so that the resulting cycle is the (k - l)th power of a Hamiltonian cycle. 4 J
  • 2. ,------ ------------------------------------, I ~ ~ ~ I ~ ~ ~ ~ ~ ~ ~ I I I I I ' I I ' I I I ~ ~j I, 'II I I ~ ~I I d ~ I II I ! ~ Dl i II i~ I D I ( I:I II'I j I I I ( ~ '~ I ~I - I ~ ~ ~ ~! I I ~'~ ~; I, I ~i j I I ~. I c Figure 1: Non-Extremal Case: Dashed lines represent the (k- 1)-paths constructed using the Connecting Lemma. Extremal Case: When our graph G is a-extremal, we use a relatively simple Konig-Hall type argument to find the (k - l)th power of a Hamiltonian cycle in G. We deal with the extremal case in Section~- 3. Main Tools We shall assume that n is s~ciently large and use the following main pa- rameters: O~«a«l, (3) where a « b means that a is sufficiently small compared to b. In order to present the results transparently we do not compute the actual dependencies, although it could be done. 3.1. Complete k-Partite Subgraphs -~ (20] H Reg&laLiey L@ffiiha [2UJ w& Meu to PiO ths Rosa 8 9 near con- .ii(tr s; I Stidd , here ::ve ([§6 iii& ' hmentesy methods U§jpg OM' the Bol- lobas-Erdos-Simonovits....,,wa• .,..,,. e•If Lemma 2 (Theorem 3.1 on page 328 in [1]). There is an absolute constant 5