Paths of specified length in random k-partite graphs

C.R. Subramanian (1)

1 CSA - Department of Computer Science and Automation [Bangalore]

Fix positive integers k and l. Consider a random k-partite graph on n vertices obtained by partitioning the vertex set into V_i, (i=1, \ldots,k) each having size Ω (n) and choosing each possible edge with probability p. Consider any vertex x in any V_i and any vertex y. We show that the expected number of simple paths of even length l between x and y differ significantly depending on whether y belongs to the same V_i (as x does) or not. A similar phenomenon occurs when l is odd. This result holds even when k,l vary slowly with n. This fact has implications to coloring random graphs. The proof is based on establishing bijections between sets of paths.

Mots clés: random graphs; paths; bijections

Domaines: Mathématique discrète [cs.DM]

C.R. Subramanian. Paths of specified length in random k-partite graphs. Discrete Mathematics and Theoretical Computer Science, 2001, Vol. 4 no. 2 (2), pp.133-138. ⟨10.46298/dmtcs.286⟩. ⟨hal-00958953⟩