Abstract
A celebrated result of Chvátal, Rödl, Szemerédi and Trotter states (in slightly weakened form) that, for every natural number Δ, there is a constant r Δ such that, for any connected n-vertex graph G with maximum degree Δ, the Ramsey number R(G,G) is at most r Δ n, provided n is sufficiently large.
In 1987, Burr made a strong conjecture implying that one may take r Δ = Δ. However, Graham, Rödl and Ruciński showed, by taking G to be a suitable expander graph, that necessarily r Δ > 2cΔ for some constant c>0. We show that the use of expanders is essential: if we impose the additional restriction that the bandwidth of G be at most some function β(n)=o(n), then R(G,G)≤(2χ(G)+4)n≤(2Δ+6)n, i.e., r Δ =2Δ+6 suffices. On the other hand, we show that Burr’s conjecture itself fails even for P k n , the kth power of a path P n .
Brandt showed that for any c, if Δ is sufficiently large, there are connected n-vertex graphs G with Δ(G)≤Δ but R(G,K 3) > cn. We show that, given Δ and H, there are β>0 and n 0 such that, if G is a connected graph on n≥n 0 vertices with maximum degree at most Δ and bandwidth at most β n , then we have R(G,H)=(χ(H)−1)(n−1)+σ(H), where σ(H) is the smallest size of any part in any χ(H)-partition of H. We also show that the same conclusion holds without any restriction on the maximum degree of G if the bandwidth of G is at most ɛ(H) log n=log logn.
Similar content being viewed by others
References
P. Allen: Covering two-edge-coloured complete graphs with two disjoint monochromatic cycles, Combin. Probab. Comput. 17 (2008), 471–486.
K. Appel and W. Haken: Every planar map is four colorable. Part I. Discharging, Illinois J. Math. 21 (1977), 429–490.
K. Appel, W. Haken and J. Koch: Every planar map is four colorable. Part II. Reducibility, Illinois J. Math. 21 (1977), 491–567.
J. A. Bondy, P. Erdős: Ramsey numbers for cycles in graphs, J. Combinatorial Theory Ser. B 14 (1973), 46–54.
J. Böttcher, K. Pruessman, A. Taraz and A. Würfl: Bandwidth, expansion, treewidth, separators, and universality for bounded degree graphs, Eur. J. Comb. 31 (2010), 1217–1227.
J. Böttcher, M. Schacht and A. Taraz: Proof of the bandwidth conjecture of Bollobás and Komlós, Mathematische Annalen 343(1) (2009), 175–205.
S. Brandt: Expanding graphs and Ramsey numbers, available at Freie Universitäat, Berlin preprint server, ftp://ftp.math.fu-berlin.de/pub/math/publ/pre/1996/pr-a-96-24.ps (1996).
W. G. Brown: On graphs that do not contain a Thomsen graph, Canad. Math. Bull. 9 (1966), 281–285.
S. A. Burr: Ramsey numbers involving graphs with long suspended paths, J. London Math. Soc. 24 (1981), 405–413.
S. A. Burr: What can we hope to accomplish in generalized Ramsey theory?, Discrete Math. 67 (1987), 215–225.
S. A. Burr and P. Erdős: Generalizations of a Ramsey-theoretic result of Chvátal, J. Graph Theory 7 (1983), 39–51.
S. A. Burr, P. Erdős, R. J. Faudree, C. C. Rousseau, R. H. Schelp: The Ramsey number for the pair complete bipartite graph-graph of limited degree, Graph theory with applications to algorithms and computer science (Kalamazoo, Mich., 1984), 163–174, Wiley, New York, 1985.
G. Chen and R. H. Schelp: Graphs with linearly bounded Ramsey numbers, J. Combin. Theory Ser. B 57 (1993), 138–149.
V. Chvátal: Tree-complete graph Ramsey number, J. Graph Theory 1 (1977), 93.
V. Chvátal and F. Harary: Generalized Ramsey theory for graphs, III. Small off-diagonal numbers, Pacific J. Math 41 (1972), 335–345.
V. Chvátal, V. Rödl, E. Szemerédi and W. T. Trotter: The Ramsey number of a graph with a bounded maximum degree, J. Combin. Theory Ser. B 34 (1983) 239–243.
D. Conlon: Hypergraph packing and sparse bipartite Ramsey numbers, Combin. Probab. Comput. 18 (2009), 913–923.
D. Conlon, J. Fox and B. Sudakov: On two problems in graph Ramsey theory, Combinatorica 32 (2012), 513–535.
R. Diestel: Graph theory, third ed., Graduate Texts in Mathematics, vol. 173, Springer-Verlag, Berlin, 2005.
P. Erdős: Some remarks on the theory of graphs, Bull. Amer. Math. Soc 53 (1947), 292–294.
P. Erdős, R. J. Faudree, C. C. Rousseau and R. H. Schelp: On cycle-complete graph Ramsey numbers, J. Graph Theory 2 (1978), 53–64.
P. Erdős and T. Gallai: On maximal paths and circuits of graphs, Acta Math. Acad. Sci. Hungar. 10 (1959), 337–356.
P. Erdős and G. Szekeres: A combinatorial problem in geometry, Composito Math. 2 (1935), 464–470.
R. J. Faudree and R. H. Schelp: All Ramsey numbers for cycles in graphs, Discrete Math. 8 (1974), 313–329.
J. Fox and B. Sudakov: Density theorems for bipartite graphs and related Ramseytype results, Combinatorica 29 (2009), 153–196.
L. Gerencsér and A. Gyárfás: On Ramsey-type problems, Annales Universitatis Scientiarum Budapestinensis, Eötvös Sect. Math. 10 (1967), 167–170.
R. L. Graham, V. Rödl and A. Ruciński: On graphs with linear Ramsey numbers, J. Graph Theory 35 (2000), 176–192.
H. A. Kierstead and W. T. Trotter: Planar graph coloring with an uncooperative partner, J. Graph Theory 18 (1994), 569–584.
Y. Kohayakawa, M. Simonovits and J. Skokan: The 3-coloured Ramsey number of odd cycles, J. Combin. Theory Ser. B, to appear.
Y. Kohayakawa, M. Simonovits and J. Skokan: Stability of Ramsey numbers for cycles, manuscript, 2008.
J. Komlós, G. N. Sárközy and E. Szemerédi: Blow-up lemma, Combinatorica 17 (1997), 109–123.
T. Kővári, V. Sós and P. Turán: On a problem of K. Zarankiewicz, Colloquium Math. 3 (1954), 50–57.
V. Nikiforov: The cycle-complete graph Ramsey numbers, Combin. Probab. Comput. 14 (2005), 349–370.
V. Nikiforov and C. C. Rousseau: Ramsey goodness and beyond, Combinatorica 29 (2009), 227–262.
S. P. Radziszowski: Small Ramsey numbers, Electronic J. Combin DS1 (2006), 60pp.
V. Rosta: On a Ramsey-type problem of J. A. Bondy and P. Erdős. I, II, J. Combin. Theory Ser. B 15 (1973), 94–105 and 105–120.
G. Sárközy, M. Schacht and A. Taraz: Two and three colour Ramsey numbers for bipartite graphs with small bandwidth, in preparation.
N. Sauer and J. Spencer: Edge disjoint placement of graphs, J. Combin. Theory Ser. B 25 (1978), 295–302.
J. Spencer: Asymptotic lower bounds for Ramsey functions, Discrete Math. 20 (1977), 69–76.
E. Szemerédi: Regular partitions of graphs, Colloques Internationaux C.N.R.S. Vol. 260, in Problémes Combinatoires et Théorie des Graphes, Orsay (1976), 399–401.
Author information
Authors and Affiliations
Corresponding author
Additional information
During this work PA was supported by DIMAP and Mathematics Institute, University of Warwick, U.K., EPSRC award EP/D063191/1.