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Improving a result of Erd˝ os, Gyárfás and Pyber for large n we show that for every integer r 2 there exists a constant n 0 = n 0 (r) such that if n n 0 and the edges of the complete graph K n are colored with r colors then the vertex set of K n can be partitioned into at most 100r log r vertex disjoint monochromatic cycles.
Let f d (G) denote the minimum number of edges that have to be added to a graph G to transform it into a graph of diameter at most d. We prove that for any graph G with maximum degree D and n > n 0 (D) vertices, f 2 (G) = n − D − 1 and f 3 (G) ≥ n − O(D 3). For d ≥ 4, f d (G) depends strongly on the actual structure of G, not only on the maximum degree of(More)
The zero-error capacity region of r-out-of-T user multiple-access OR channel is investigated. A family F of subsets of [n] = {1, ..., n} is an r-single-user-tracing superimposed code (r-SUT) if there exists such a single-user-tracing function /spl phi/:2/sup [n]/ /spl rarr/ F that for all F' /spl sube/ F with 1 /spl les/ |F'| /spl les/ r, /spl phi/(/spl(More)
A family of n-dimensional unit norm vectors is a Euclidean superimposed code, if the sums of any two distinct at most m-tuples of vectors are separated by a certain minimum Euclidean distance d. Ericson and Györfi [8] proved that the rate of such a code is between (log m)/4m and (log m)/m for m large enough. In this paper – improving the above longstanding(More)
In this paper we deal with codes identifying sets of vertices in random graphs, that is l-identifying codes. These codes enable us to detect sets of faulty processors in a multiprocessor system, assuming that the maximum number of faulty processors is bounded by a fixed constant l. The l-identifying codes or simply identifying codes are of special interest.(More)