Abstract
Define a k-minimum-difference-representation (k-MDR) of a graph G to be a family of sets \({\{S(v): v\in V(G)\}}\) such that u and v are adjacent in G if and only if min{|S(u)−S(v)|, |S(v)−S(u)|} ≥ k. Define ρ min(G) to be the smallest k for which G has a k-MDR. In this note, we show that {ρ min(G)} is unbounded. In particular, we prove that for every k there is an n 0 such that for n > n 0 ‘almost all’ graphs of order n satisfy ρ min(G) > k. As our main tool, we prove a Ramsey-type result on traces of hypergraphs.
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References
Axenovich M., Balogh J.: Graphs having small number of sizes on induced k-subgraphs. SIAM J. Discrete Math. 21, 264–272 (2007)
Balogh J., Bollobás B.: Unavoidable traces of set systems. Combinatorica 25(6), 633–643 (2005)
Balogh J., Bollobás B., Saks M., Sós V.T.: On the diversity function of a hereditary graph property. J. Combin. Theory Ser. B 99, 9–19 (2009)
Balogh J., Bollobás B., Weinreich D.: A jump to the Bell number for hereditary graph properties. J. Combin. Theory Ser. B. 95(1), 29–48 (2005)
Balogh J., Pluhár A.: A sharp edge bound on the interval number of a graph. J. Graph Theory 32(2), 153–159 (1999)
Balogh J., Keevash P., Sudakov B.: Disjoint representability of sets and their complements. J. Combin. Theory Ser. B 95(1), 12–28 (2005)
Boros E., Gurvich V., Meshulam R.: Difference Graphs. Discrete Math. 276(1–3), 59–64 (2004)
Chang, Y.-W., Hutchinson, J.P., Jacobson, M.S., Lehel, J., West, D.B.: The bar visibility number of a graph, SIAM J. Discrete Math. 18 (3), 462–471 (2004/2005)
Füredi, Z.: Talk at 2006 Fall Southeastern Section Meeting Fayetteville, AR, 3–4 November. Meeting # 1022. http://www.ams.org/amsmtgs/2127abstracts/1022-05-153.pdf (2006)
Füredi, Z.: Personal communication (2006)
Lovász L., Nešetřil J., Pultr A.: On a product dimension of graphs. J. Combin. Theory Ser. B 29(1), 47–67 (1980)
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Research supported in part by NSF grants DMS-0302804, DMS-0603769 and DMS-0600303, UIUC Campus Research Board 06139 and 07048, and OTKA 049398. Research supported in part by UIUC Campus Research Board 07048.
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Balogh, J., Prince, N. Minimum Difference Representations of Graphs. Graphs and Combinatorics 25, 647–655 (2009). https://doi.org/10.1007/s00373-010-0875-3
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DOI: https://doi.org/10.1007/s00373-010-0875-3