Existence of a spanning cyclic ladder graph in a balanced
Transcription
Existence of a spanning cyclic ladder graph in a balanced
Existence of a spanning cyclic ladder graph in a balanced bipartite graph Peter Heinig Introduction The Bachelor's Thesis(continued) It is common in mathematics that rushing in does not succeed. If, to sketch an example, This leads to the (as yet unproved) conjecture that there is the following f : D !C is a holomorphic function on an open disk D the boundary of is not D C , and every point on a singular point (that means that around every point of there is an open disk and a holomorphic continuation of holomorphically extended to an open disk ~ D f to that disk), then f strengthening in the case of bipartite balanced graphs. Note that the @D minimum degree in the hypothesis has been halved. > 0 and every 2 N there exists = (; ) > 0 and n0 = n0(; ) such that every balanced bipartite 1 2 1 1 graph G with jG j n0 and (G ) 2 2 + jG j = 4 + jG j contains as a subgraph every balanced bipartite graph H with jH j = jG j and (H ) and bw(H ) jH j. G My result proves a special case of this conjecture because H := CL 2 not only has the proportionally small bandwidth of at most jH j but r even constant bandwidth, since 8r 2 N4 : bw CL = 4. can be Conjecture. For every having the same midpoint and a strictly D, but it is not possible to prove this in general by greedily applying the local continuation property to run around the circle @D , since in general there is no control over the radii of the disks around the points of @D . However, it is easy to larger radius than prove it by arguing in an appropriate and more global way (an example can be found in [2], Theorem 2.19.). j Analogously, there is a deep global principle in modern combinatorics which in its formulation for graphs essentially states that for literally every graph, if only it is large enough (the numbers of vertices that are known to be sucient turn out to be extremely large but they can be explicitly stated, albeit normally not in decimal j The second way is to view the result as a generalization of the fact notation), certain global preparatory provisions can be made, which allow one to prove statements (mostly about the existence of certain subgraphs) which otherwise seem (which is known at least since 1963, see [1]) that for a balanced bipartite impossible to prove by greedily and locally appealing to the given hypotheses, these graph hypotheses often being of a very local kind, like minimum degree conditions. The most G jG j + 1 implies the existence of a minimum degree of at least 4 +1 term is replaced a Hamilton cycle. In the case of my theorem, the important keyword here is Szemerédi's regularity lemma , about which more can be by a learned in sources like [9],[8],[3],[4],[10],[11],[12]. + jG j term, >0 being an arbitrarily small constant, i.e. the excess minimum degree is allowed to be arbitrarily small To reiterate a point from above, one of the most interesting aspects of the regularity but gets arbitrarily large lemma is that there are several facts whose formulation looks quite clean and far Gj CL 2 relative to jG j absolutely . In return for this stronger hypothesis j removed from the rather technical statements concomitant with the regularity lemma, the subgraph but no one has a clue how to prove them without it. One example is the proof of a Hamilton cycle. For example, by counting the Hamilton cycles that a conjecture of Seymour that a graph with least k n k +1 must contain as a subgraph the n Gj CL 2 alone contains, regardless of the many extra edges thehost graph G has got, it follows that a minimum degree of at least 14 + jG j implies (when jG j is large enough) not only the existence of one Hamilton cycle jG j Hamilton cycles if jG j 2 ( mod 4 ) and of jG j + 2 but of 2 2 Hamilton cycles if jG j 0 ( mod 4 ), i.e. a non-constant number of j vertices and a minimum degree of at k th power of a Hamilton cycle (see [6] for more). The Bachelor's Thesis distinct Hamilton cycles. The main result of my thesis, which was supervised by Anusch Taraz Note that these minimum degree conditions are very near to being and Julia Böttcher, is a proof for the following statement. > 0 there exists n0 = n0( ) 2 N such that every balanced bipartite graph G with jG j n0 and (G ) 1 + jG j contains CL G2 as a subgraph. 4 Here, jG j denotes the number of vertices of G , being `balanced' means best possible since a minimum degree of G Theorem [Heinig 2008]. For every j CL Gj 2 is dened to be K 2 Gj C2 j two vertices, jG j 4 does not even imply that is connected (why?), let alone that it contains a Hamilton cycle. j A glance at the proof that the two classes of the bipartite graph have equal cardinality, and j being forced to appear is much more structured than a cycle Gj CL 2 j Gj The gure below is a schematic overview of how C 2 where K 2 denotes a complete graph with jG j vertices, and the cartesian product with 2 j into G. First G gets embedded is globally prepared using the regularity lemma (and Gj CL 2 j much additional tweaking), then gets embedded at one fell swoop This picture shows a using the so-called Blow-Up Lemma (see [5]). The diculty mostly CL 2 in of jG j = lies in (1) the fact that the regularity lemma method invariably involves Gj j drawing the 64, der of case ignoring rungs. It ample for with 32 an ex- is relatively tiny but absolutely really prove the existence of a subgraph which has large number of exactly as many vertices as the graph it exists in, and (2) the minimum degree condition type which of graph whose exis- would tence as a spanning tttttttttttttttttttttttttttttttttttttttttttt x x x x x x x x x x subgraph is proved in tttttttttttttttttttttttttttttttttttttttttttt my thesis. tttttttttttttttttttttttttttttttttttttttttttt the arbitrarily waste vertices which have to be re-distributed skillfully in order to i.e. a cyclic ladgraph a 11 12 is barely not 1s1 above be a threshold sucient. 21 22 2s2 31 32 which (as y1 1 y1s 1 y1 s1 −1 y2 1 y2s 2 y2 s2 −1 mentioned) tttttttttttttttttttttttttttttttttttttttttttt x3s 3 41 42 tttttttttttttttttttttttttttttttttttttttttttt of graphs. already y3 1 y3s 3 y3 s3 −1 y4 1 tttttttttttttttttttttttttttttttttttttttttttt x4s 4 x` x` 1 2 y4s 4 y4 s4 −1 x` y` 1 s` x1 1 x1 2 y` s` y` s` −1 There are two good ways to put this result in context and the rst of them will be outlined now. In [13] the authors prove the following theorem, where (G ) bandwidth of the graph denotes the chromatic number and bw Selected References (G ) the G (standard notions whose denitions are easily [1] John Moon, Leo Moser: On hamiltonian bipartite graphs. Israel Journal of Mathematics 1: 163-165 (1963) [2] Aleksei Ivanovich Markushevich: Theory of Functions of a Complex Variable, Volume 1, Revised English Edition, Translated and Edited by Richard A. Silverman, Prentice-Hall, Inc., 11th printing (1966) [3] Miklós Simonovits, Vera T. Sós: Szemerédi's Partition and Quasirandomness. Random Structures and Algorithms, Vol. 2, No. 1: 1-10 (1991) [4] Noga Alon, Richard Alter Duke, Hanno Lefmann, Vojtech Rödl, Raphael Yuster: The Algorithmic Aspects of the Regularity Lemma. Journal of Algorithms 16: 80-109 (1994) [5] János Komlós, Gábor Sárközy, Endre Szemerédi: Blow-Up Lemma. Combinatorica 17: 109-123 (1997) [6] János Komlós, Gábor N. Sárközy, Endre Szemerédi: Proof of the Seymour conjecture for Large Graphs. Annals of Combinatorics, Volume 2, pp. 42-60 (1998) [7] Sarmad Abbasi: How Tight is the Bollobás-Komlós Conjecture? Graphs and Combinatorics 16: 129-137 (2000) [8] Reinhard Diestel: Graph Theory. Springer-Verlag, Graduate Texts in Mathematics, Volume 173, Third Edition (2005) [9] Jozef Skokan: Regularity Lemma Basics. (not published in printed form; easily found on the web)(2006) [10] Terence Tao: Szemerédi's Regularity Lemma revisited. Contributions to Discrete Mathematics 1, 8-28 (2006) [11] Ben Joseph Green, Terence Tao: Szemerédi's Theorem. Scholarpedia, 2(7):3446 (2007) [12] Terence Tao: What is good mathematics? Bulletin (New Series) of The American Mathematical Society, Volume 4, Number 4 : p. 628-634 (2007) [13] Julia Böttcher, Mathias Schacht, Anusch Taraz: Proof of the bandwidth conjecture of Bollobás and Komlós. Mathematische Annalen, Volume 343, Number 1, Pages 175-205 (2009) found). > 0 and for every r 2 N and for every 2 N there exists = (; r; ) > 0 and n0 = n0 (; r; ) such that every r 1 graph G with jG j n0 and (G ) r + jG j contains as a subgraph every graph H with jH j = jG j and (H ) r and (H ) and bw(H ) jH j. Theorem. For every [13] Julia Böttcher, Mathias Schacht, Anusch Taraz: Chair of Applied Geometry and Discrete Mathematics (M9) j Boltzmannstr. 3 j 85748 Garching j www-m9.ma.tum.de