Clique-width III: Hamiltonian cycle and the odd case of graph coloring

Max-Cut, Edge Dominating Set, Graph Coloring, and Hamiltonian Cycle on graphs of bounded cliquewidth have received significant attention as they can be formulated inMSO₂ (and, therefore, have linear-time algorithms on bounded treewidth graphs by the celebrated Courcelle's theorem), but cannot be formulated in MSO1 (which would have yielded linear-time algorithms on bounded clique-width graphs by a well-known theorem of Courcelle, Makowsky, and Rotics). Each of these problems can be solved in time g(k)n^f(k) on graphs of clique-width k. Fomin et al. (2010) showed that the running times cannot be improved to g(k)n^O(1) assuming W[1]≠FPT. However, this does not rule out non-trivial improvements to the exponent f (k) in the running times. In a follow-up paper, Fomin et al. (2014) improved the running times for EdgeDominating Set and Max-Cut to n ^O(k) , and proved that these problems cannot be solved in time g(k)n^o(k) unless ETH fails. Thus, prior to this work, Edge Dominating Set and Max-Cut were known to have tight n^θ(k) algorithmic upper and lower bounds. In this article, we provide lower bounds for Hamiltonian Cycle and Graph Coloring. For Hamiltonian Cycle, our lower bound g(k)n^o(k) matches asymptotically the recent upper bound n^O(k) due to Bergougnoux, Kanté, and Kwon (2017). As opposed to the asymptotically tight n^θ(k) bounds for Edge Dominating Set, Max-Cut, and Hamiltonian Cycle, the Graph Coloring problem has an upper bound of n^{O(2k )} and a lower bound of merely n^{o( 4vk)} (implicit from the W[1]-hardness proof). In this article, we close the gap for Graph Coloring by proving a lower bound of n^{2 o(k)}. This shows that Graph Coloring behaves qualitatively different from the other three problems. To the best of our knowledge, Graph Coloring is the first natural problem known to require exponential dependence on the parameter in the exponent of n.