On the dynamic width of the 3-colorability problem

A graphG is 3-colorable if and only if it maps homomorphically to the complete 3-vertex graph K3. The last condition can be checked by a k-consistency algorithm where the parameter k has to be chosen large enough, dependent on G. Let W (G) denote the minimum k sufficient for this purpose. For a non3-colorable graph G, W (G) is equal to the minimum k such that G can be distinguished from K3 in the k-variable existential-positive first-order logic. We define the dynamic width of the 3-colorability problem as the function W (n) = maxGW (G), where the maximum is taken over all non-3-colorable G with n vertices. The assumption NP 6= P implies that W (n) is unbounded. Indeed, a lower bound W (n) = Ω(log log n/ log log log n) follows unconditionally from the work of Nešetřil and Zhu [26] on bounded treewidth duality. The Exponential Time Hypothesis implies a much stronger bound W (n) = Ω(n/ log n) and indeed we unconditionally prove that W (n) = Ω(n). In fact, an even stronger statement is true: A first-order sentence distinguishing any 3-colorable graph on n vertices from any non-3-colorable graph on n vertices must have Ω(n) variables. On the other hand, we observe that W (G) ≤ 3α(G) + 1 and W (G) ≤ n − α(G) + 1 for every non-3-colorable graph G with n vertices, where α(G) denotes the independence number of G. This implies that W (n) ≤ 34 n + 1, improving on the trivial upper bound W (n) ≤ n. We also show that W (G) > 1 16 g(G) for every non-3-colorable graph G, where g(G) denotes the girth of G. Finally, we consider the function W (n) over planar graphs and prove that W (n) = Θ( √ n) in the case. Supported by EPSRC grant EP/H026835. Research visit to Germany supported by DAAD grant A/13/05456. Supported by DFG grant VE 652/1–1. On leave from the Institute for Applied Problems of Mechanics and Mathematics, Lviv, Ukraine.