Does the following holds?
For every bipartite graph $H$ and every graph $G$ with $\frac{\Delta(G)}{\delta(G)}\leq 2$, $$t(H,G)\geq t(K_2, G)^{e(H)}.$$
If not sure, is this a equal question as Sidorenko's conjecture or a subquestion of Sidorenko?
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$\begingroup$ What does $t(H,G)$ mean? $\endgroup$– bofCommented May 1 at 18:14
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$\begingroup$ @bof Take a random onto map f from V(H) to V(G), the prob. that this map keep adjacency, i.e. a~b implies f(a)~f(b). If G's adjacency matrix's every element is 1, then for every H t(H,G)=V(G)^V(H)/V(G)^V(H)=1. $\endgroup$– tom jerryCommented May 2 at 8:51
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1 Answer
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Proving the conjecture for host graphs $G$ with $\Delta/\delta$ bounded is equivalent to proving the full conjecture. One may even assume that the host graph $G$ is both vertex- and edge-transitive, a result of Szegedy that can be found at https://arxiv.org/abs/1504.00858.
