Partitioning 2-edge-colored graphs by monochromatic paths and cycles

Abstract

We present results on partitioning the vertices of 2-edge-colored graphs into monochromatic paths and cycles. We prove asymptotically the two-color case of a conjecture of Sárközy: the vertex set of every 2-edge-colored graph can be partitioned into at most 2α(G) monochromatic cycles, where α(G) denotes the independence number of G. Another direction, emerged recently from a conjecture of Schelp, is to consider colorings of graphs with given minimum degree. We prove that apart from o(|V (G)|) vertices, the vertex set of any 2-edge-colored graph G with minimum degree at least \(\tfrac{{(1 + \varepsilon )3|V(G)|}} {4}\)can be covered by the vertices of two vertex disjoint monochromatic cycles of distinct colors. Finally, under the assumption that \(\bar G\) does not contain a fixed bipartite graph H, we show that in every 2-edge-coloring of G, |V (G)| − c(H) vertices can be covered by two vertex disjoint paths of different colors, where c(H) is a constant depending only on H. In particular, we prove that c(C 4)=1, which is best possible.

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Correspondence to József Balogh.

Additional information

Research supported in part by NSF CAREER Grant DMS-0745185, UIUC Campus Research Board Grant 11067, and OTKA Grant K 76099.

Research is supported by OTKA Grants PD 75837 and K 76099, and the János Bolyai Research Scholarship of the Hungarian Academy of Sciences. Present address: School of Mathematical Sciences, Monash University, 3800 Victoria.

Research supported by Hungarian Science Foundation Grant OTKA NN 102029, under the EuroGIGA programs ComPoSe and GraDR.

Research supported in part by OTKA Grant K104373.

Research supported in part by NSF Grant DMS-0968699 and by OTKA Grant K104373.

Part of the research reported in this paper was done at the 3rd Emléktábla Workshop (2011) in Balatonalmádi, Hungary.

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Balogh, J., Barát, J., Gerbner, D. et al. Partitioning 2-edge-colored graphs by monochromatic paths and cycles. Combinatorica 34, 507–526 (2014). https://doi.org/10.1007/s00493-014-2935-4

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Mathematics Subject Classification (2000)

  • 05C15
  • 05D10