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BEBERAPA KELAS GRAF RAMSEY MINIMAL UNTUK LINTASAN P_3 VERSUS P_5
In 1930, Frank Plumpton Ramsey has introduced Ramsey's theory, in his paper titled On a Problem of Formal Logic. This study became morepopular since Erdős and Szekeres applied Ramsey's theory to graph theory. Suppose given the graph F, G and H. The notation F → (G, H) states thatfor any red-blue coloring of the edges of F implies F containing a red subgraph of G or a blue subgraph of H. The graph F is said to be the Ramsey graph for graph G versus H (pair G and H) if F → (G, H). Graph F is called Ramsey minimal graph for G versus H if first, F → (G, H) and second, F satisfies the minimality property i.e. for each e ∈ E (F), then F-e ↛ (G, H). The class of all Ramsey (G, H) minimal graphs is denoted by (G, H). The class (G, H) is called Ramsey infinite or finite if (G, H) is infinite or finite, respectively. The study about Ramsey minimal graph is still continuously being developed and examined, although in general it is not easy to characterize or determine the graphs included in the (G, H), especially if (G, H) is an infinite Ramsey class. The characterization of graphs in (, ) has been obtained. However, the characterization of graphs in (, ), for every 3 ≤ m < n is still open. In this article, we will determine some infinite classes of Ramsey minimal graphs for paths versus .
Directed Ramsey number for trees
In this paper, we study Ramsey-type problems for directed graphs. We first
consider the -colour oriented Ramsey number of , denoted by
, which is the least for which every
-edge-coloured tournament on vertices contains a monochromatic copy of
. We prove that for any oriented
tree . This is a generalisation of a similar result for directed paths by
Chv\'atal and by Gy\'arf\'as and Lehel, and answers a question of Yuster. In
general, it is tight up to a constant factor.
We also consider the -colour directed Ramsey number
of , which is defined as above, but, instead
of colouring tournaments, we colour the complete directed graph of order .
Here we show that for any
oriented tree , which is again tight up to a constant factor, and it
generalises a result by Williamson and by Gy\'arf\'as and Lehel who determined
the -colour directed Ramsey number of directed paths.Comment: 27 pages, 14 figure
Ramsey numbers of ordered graphs
An ordered graph is a pair where is a graph and
is a total ordering of its vertices. The ordered Ramsey number
is the minimum number such that every ordered
complete graph with vertices and with edges colored by two colors contains
a monochromatic copy of .
In contrast with the case of unordered graphs, we show that there are
arbitrarily large ordered matchings on vertices for which
is superpolynomial in . This implies that
ordered Ramsey numbers of the same graph can grow superpolynomially in the size
of the graph in one ordering and remain linear in another ordering.
We also prove that the ordered Ramsey number is
polynomial in the number of vertices of if the bandwidth of
is constant or if is an ordered graph of constant
degeneracy and constant interval chromatic number. The first result gives a
positive answer to a question of Conlon, Fox, Lee, and Sudakov.
For a few special classes of ordered paths, stars or matchings, we give
asymptotically tight bounds on their ordered Ramsey numbers. For so-called
monotone cycles we compute their ordered Ramsey numbers exactly. This result
implies exact formulas for geometric Ramsey numbers of cycles introduced by
K\'arolyi, Pach, T\'oth, and Valtr.Comment: 29 pages, 13 figures, to appear in Electronic Journal of
Combinatoric
The random graph
Erd\H{o}s and R\'{e}nyi showed the paradoxical result that there is a unique
(and highly symmetric) countably infinite random graph. This graph, and its
automorphism group, form the subject of the present survey.Comment: Revised chapter for new edition of book "The Mathematics of Paul
Erd\H{o}s
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