By Rudenskaya O. G.
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Prove that there are at most 3n pairs of these points at distance exactly one. Suppose G is a connected graph with k edges. , which is of degree at least two), the greatest common divisor of the integers labeling those edges is 1. (32nd IMO, 1991/4) Nine mathematicians meet at an international conference and discover that among any three of them, at least two speak a common language. If each of the mathematicians can speak at most three page 47 January 19, 2015 48 15:48 BC: 9426 – Graph Theory: Undergraduate Mathematics Book Graph Theory languages, prove that there are at least three of the mathematicians who can speak the same language.
7. Is your answer for (iii) double your answer for (i)? Is your answer for (iv) an even number? Construct a (6, 7)-multigraph in which every vertex is odd. , vn }. , n. 16. Let B(G)T denote the transpose of B(G). Find B(G)B(G)T and give an interpretation for the (i, j)-entry in B(G)B(G)T . Let G be a (8, 15)-graph in which each vertex is of degree 3 or 5. How many vertices of degree 5 does G have? Construct one such graph G. Let H be a graph of order 10 such that 3 ≤ d(v) ≤ 5 for each vertex v in H.
Being even or odd) of their degrees. A vertex w in G is said to be even if d(w) is even; and said to be odd if d(w) is odd. Thus, in the multigraph G of Fig. 2, there are five even vertices: b, j, p, w and y; and four odd vertices: a, c, x and z. 8. (a) How many odd vertices are there in each of the multigraphs shown in the previous examples? (b) Can you construct a multigraph containing (i) exactly one odd vertex? (ii) exactly three odd vertices? Instead of merely considering the multigraph of Fig.