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This is a rather rare sort of result in Ramsey theory. 57 THEOREM If n. , n2, . . , nc = max (nt , . . , nc), then r(n1 K2, n2 K2 , . . , nc K2) ( Cockayne and Lorimer are positive = integers and n1 c n1 + I + � (nt - I) i= l l975a) . Here is the way to construct an example showing that less than the above <" -•> into sets V;, i = 1 , 2, value will not do: Partition the vertices of K, + �c �r-1 . . , c, so that I V. I = 2nt - I and I V;I = n; - I for i 2, 3, . . , c. Color with the first color all edges which are incident with two vertices in V..

Tutte ( 196 1) . G is 3-connected if and only if G is a wheel or can be obtained from a wheel by a finite sequence of operations of the following types: ( a) The addition of a line. ( b) Replacing a point v, having d(v) � 4, by two adjacent points u and v in such a way that each point formerly adjacent to v is adjacent to exactly one of u and w, and in the resulting graph d(u) � 3 and d(w) � 3. We refer to this operation as a split. Neither operation by itself characterizes 3-connected graphs. To see that operations of type ( a) alone do not suffice, consider the prism which has Cp as base.

We conclude the chapter with an example on the line core of a graph. The line core of a graph G is the subgraph of G induced by the union of all independent sets I of lines, if any, that contain a0( G ) points. 26 Not every graph has a line core. Let G = C2n+I· Then /31 (G ) = n < n + l = a0( G ). Hence, G can have no set I of independent lines containing a0(G ) points (Dulmage and Mendelsohn 1958). 38 Chapter 4 Extremal Problems 1. RAMSEY NUMBERS Most of the examples in this chapter deal with Ramsey numbers, and we have collected them together in this special section.

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