Short Answers to Exponentially Long Questions: Extremal Aspects of Homomorphism Duality

Authors:
Jaroslav Nesetril;Claude Tardif
Affiliations:
-;-
Venue:
SIAM Journal on Discrete Mathematics
Year:
2005

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Abstract

We prove that there exists a constant $k$ such that for every $n \geq 1$ there exists a directed core graph $H_n$ with at least $2^n$ vertices such that a directed graph $G$ is $H_n$-colorable if and only if every subgraph of $G$ with at most $kn\log(n)$ vertices is $H_n$-colorable. Our examples show that in general the "duals of relational structures" in the sense of [J. Nesetril and C. Tardif, J. Combin. Theory Ser. B, 80 (2000), pp. 80-97] can have superpolynomial size. The construction given in this paper gives a double exponential upper bound for such a construction. Here we improve this to an exponential upper bound.