TY - JOUR

T1 - Graph powers, Delsarte, Hoffman, Ramsey, and Shannon

AU - Alon, Noga

AU - Lubetzky, Eyal

PY - 2007

Y1 - 2007

N2 - The kth p-power of a graph G is the graph on the vertex set V(G) k, where two k-tuples are adjacent iff the number of their coordinates which are adjacent in G is not congruent to 0 modulo p. The clique number of powers of G is polylogarithmic in the number of vertices; thus graphs with small independence numbers in their p-powers do not contain large homogeneous subsets. We provide algebraic upper bounds for the asymptotic behavior of independence numbers of such powers, settling a conjecture of [N. Alon and E. Lubetzky, Combinatorica, 27 (2007), pp. 13-33] up to a factor of 2. For precise bounds on some graphs, we apply Delsarte's linear programming bound and Hoffman's eigenvalue bound. Finally, we show that for any nontrivial graph G, one can point out specific induced subgraphs of large p-powers of G with neither a large clique nor a large independent set. We prove that the larger the Shannon capacity of Ḡ is, the larger these subgraphs are, and if G is the complete graph, then some p-power of G matches the bounds of the Frankl-Wilson Ramsey construction, and is in fact a subgraph of a variant of that construction.

AB - The kth p-power of a graph G is the graph on the vertex set V(G) k, where two k-tuples are adjacent iff the number of their coordinates which are adjacent in G is not congruent to 0 modulo p. The clique number of powers of G is polylogarithmic in the number of vertices; thus graphs with small independence numbers in their p-powers do not contain large homogeneous subsets. We provide algebraic upper bounds for the asymptotic behavior of independence numbers of such powers, settling a conjecture of [N. Alon and E. Lubetzky, Combinatorica, 27 (2007), pp. 13-33] up to a factor of 2. For precise bounds on some graphs, we apply Delsarte's linear programming bound and Hoffman's eigenvalue bound. Finally, we show that for any nontrivial graph G, one can point out specific induced subgraphs of large p-powers of G with neither a large clique nor a large independent set. We prove that the larger the Shannon capacity of Ḡ is, the larger these subgraphs are, and if G is the complete graph, then some p-power of G matches the bounds of the Frankl-Wilson Ramsey construction, and is in fact a subgraph of a variant of that construction.

KW - Cliques and independent sets

KW - Delsarte's linear programming bound

KW - Eigenvalue bounds

KW - Graph powers

KW - Ramsey theory

UR - http://www.scopus.com/inward/record.url?scp=45249112834&partnerID=8YFLogxK

U2 - 10.1137/060657893

DO - 10.1137/060657893

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AN - SCOPUS:45249112834

SN - 0895-4801

VL - 21

SP - 329

EP - 348

JO - SIAM Journal on Discrete Mathematics

JF - SIAM Journal on Discrete Mathematics

IS - 2

ER -