Geodesic
Paths of specified length in random k-partite graphs
Discrete mathematics & theoretical computer science, Tome 4 (2000-2001) no. 2.

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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. 
@article{DMTCS_2001_4_2_a13,
     author = {Subramanian, C.R.},
     title = {Paths of specified length in random k-partite graphs},
     journal = {Discrete mathematics & theoretical computer science},
     publisher = {mathdoc},
     volume = {4},
     number = {2},
     year = {2000-2001},
     doi = {10.46298/dmtcs.286},
     language = {en},
     url = {http://geodesic.mathdoc.fr/articles/10.46298/dmtcs.286/}
}
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Subramanian, C.R. Paths of specified length in random k-partite graphs. Discrete mathematics & theoretical computer science, Tome 4 (2000-2001) no. 2. doi : 10.46298/dmtcs.286. http://geodesic.mathdoc.fr/articles/10.46298/dmtcs.286/

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