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Knesers conjecture, first proved by Lovsz in 1978, states that the graph with all k-element subsets of {1, 2, . . . , n} as vertices and with edges connecting disjoint sets has chromatic number n2k+2. We derive this result from Tuckers combinatorial lemma on labeling the vertices of special triangulations of the octahedral ball. By specializing a proof of Tuckers lemma, we obtain self-contained purely...
We prove a conjecture of Fredi and Mubayi: For any graph G on n vertices with minimum degree r, there exists a two-coloring of the vertices of G with colors +1 and -1, such that the closed neighborhood of each vertex contains more +1s than -1s, and altogether the number of 1s does not exceed the number of -1s by more than . As a construction by Fredi and Mubayi shows, this is asymptotically tight...
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