25 research outputs found
On Ramsey numbers of complete graphs with dropped stars
Let be the smallest integer such that for any -coloring (say,
red and blue) of the edges of , , there is either a red
copy of or a blue copy of . Let be the complete graph on
vertices from which the edges of are dropped. In this note we
present exact values for and new upper bounds
for in numerous cases. We also present some results for
the Ramsey number of Wheels versus .Comment: 9 pages ; 1 table in Discrete Applied Mathematics, Elsevier, 201
Size Ramsey Numbers Involving Double Stars and Brooms
The topics of this thesis lie in graph Ramsey theory. Given two graphs G and H, by the Ramsey theorem, there exist infinitely many graphs F such that if we partition the edges of F into two sets, say Red and Blue, then either the graph induced by the red edges contains G or the graph induced by the blue edges contains H. The minimum order of F is called the Ramsey number and the minimum of the size of F is called the size Ramsey number. They are denoted by r(G, H) and ˆr(G, H), respectively. We will investigate size Ramsey numbers involving double stars and brooms
On Size Multipartite Ramsey Numbers for Stars Versus Paths and Cycles
Let be a complete, balanced, multipartite graph consisting of partite sets and vertices in each partite set. For given two graphs and , and integer , the size multipartite Ramsey number is the smallest integer such that every factorization of the graph satisfies the following condition: either contains or contains . In 2007, Syafrizal et al. determined the size multipartite Ramsey numbers of paths versus stars, for only. Furthermore, Surahmat et al. (2014) gave the size tripartite Ramsey numbers of paths versus stars, for . In this paper, we investigate the size tripartite Ramsey numbers of paths versus stars, with all . Our results complete the previous results given by Syafrizal et al. and Surahmat et al. We also determine the size bipartite Ramsey numbers of stars versus cycles, for
Graphs with second largest eigenvalue less than
We characterize the simple connected graphs with the second largest
eigenvalue less than 1/2, which consists of 13 classes of specific graphs.
These 13 classes hint that , where is
the minimum real number for which every real number greater than is a
limit point in the set of the second largest eigenvalues of the simple
connected graphs. We leave it as a problem.Comment: 36 pages, 2 table
On tripartite common graphs
A graph H is common if the number of monochromatic copies of H in a
2-edge-colouring of the complete graph is minimised by the random colouring.
Burr and Rosta, extending a famous conjecture by Erdos, conjectured that every
graph is common. The conjectures by Erdos and by Burr and Rosta were disproved
by Thomason and by Sidorenko, respectively, in the late 1980s. Collecting new
examples for common graphs had not seen much progress since then, although very
recently, a few more graphs are verified to be common by the flag algebra
method or the recent progress on Sidorenko's conjecture.
Our contribution here is to give a new class of tripartite common graphs. The
first example class is so-called triangle-trees, which generalises two theorems
by Sidorenko and answers a question by Jagger, \v{S}\v{t}ov\'i\v{c}ek, and
Thomason from 1996. We also prove that, somewhat surprisingly, given any tree
T, there exists a triangle-tree such that the graph obtained by adding T as a
pendant tree is still common. Furthermore, we show that adding arbitrarily many
apex vertices to any connected bipartite graph on at most five vertices give a
common graph
Common graphs with arbitrary chromatic number
Ramsey's Theorem guarantees for every graph H that any 2-edge-coloring of a
sufficiently large complete graph contains a monochromatic copy of H. In 1962,
Erdos conjectured that the random 2-edge-coloring minimizes the number of
monochromatic copies of K_k, and the conjecture was extended by Burr and Rosta
to all graphs. In the late 1980s, the conjectures were disproved by Thomason
and Sidorenko, respectively. A classification of graphs whose number of
monochromatic copies is minimized by the random 2-edge-coloring, which are
referred to as common graphs, remains a challenging open problem. If
Sidorenko's Conjecture, one of the most significant open problems in extremal
graph theory, is true, then every 2-chromatic graph is common, and in fact, no
2-chromatic common graph unsettled for Sidorenko's Conjecture is known. While
examples of 3-chromatic common graphs were known for a long time, the existence
of a 4-chromatic common graph was open until 2012, and no common graph with a
larger chromatic number is known.
We construct connected k-chromatic common graphs for every k. This answers a
question posed by Hatami, Hladky, Kral, Norine and Razborov [Combin. Probab.
Comput. 21 (2012), 734-742], and a problem listed by Conlon, Fox and Sudakov
[London Math. Soc. Lecture Note Ser. 424 (2015), 49-118, Problem 2.28]. This
also answers in a stronger form the question raised by Jagger, Stovicek and
Thomason [Combinatorica 16, (1996), 123-131] whether there exists a common
graph with chromatic number at least four.Comment: Updated to include reference to arXiv:2207.0942