The authors determine the exact VC-dimension-interpolated homomorphism thresholds for cliques and prove the blowup threshold of odd cycles C_{2k-1} is 1/(2k-1).
A strengthening on consecutive odd cycles in graphs of given minimum degree
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abstract
Liu and Ma [J. Combin. Theory Ser. B, 2018] conjectured that every $2$-connected non-bipartite graph with minimum degree at least $k+1$ contains $\lceil k/2\rceil $ cycles with consecutive odd lengths. In particular, they showed that this conjecture holds when $k$ is even. In this paper, we confirm this conjecture for any $k\in \mathbb N$. Moreover, we also improve some previous results about cycles of consecutive lengths.
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Interpolating chromatic and homomorphism thresholds
The authors determine the exact VC-dimension-interpolated homomorphism thresholds for cliques and prove the blowup threshold of odd cycles C_{2k-1} is 1/(2k-1).