Restricted greedy clique decompositions and greedy clique decompositions ofK 4-free graphs

Abstract

A greedy clique decomposition of a graph is obtained by removing maximal cliques from a graph one by one until the graph is empty. It has recently been shown that any greedy clique decomposition of a graph of ordern has at mostn 2/4 cliques. In this paper, we extend this result by showing that for any positive integerp, 3≤p any clique decomposisitioof a graph of ordern obtained by removing maximal cliques of order at leastp one by one until none remain, in which case the remaining edges are removed one by one, has at mostt p-1(n) cliques. Heret p-1(n) is the number of edges in the Turán graph of ordern, which has no complete subgraphs of orderp.

In connection with greedy clique decompositions, P. Winkler conjectured that for any greedy clique decompositionC of a graphG of ordern the sum over the number of vertices in each clique ofC is at mostn 2/2. We prove this conjecture forK 4-free graphs and show that in the case of equality forC andG there are only two possibilities:

  1. (i)

    GK n/2,n/2

  2. (ii)

    G is complete 3-partite, where each part hasn/3 vertices.

We show that in either caseC is completely determined.

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McGuinness, S. Restricted greedy clique decompositions and greedy clique decompositions ofK 4-free graphs. Combinatorica 14, 321–334 (1994). https://doi.org/10.1007/BF01212980

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AMS subject classification code (1991)

  • 05 C 35