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.
Similar content being viewed by others
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)
Author information
Authors and Affiliations
Corresponding author
Additional information
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.
Rights and permissions
About this article
Cite this article
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
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00373-010-0875-3