A Generalization of Roman Domination Critical Graphs

The van der Waerden number \( W(r, k) \) is the least integer \( N \) such that every \( r \)-coloring of \( \{1, 2, \ldots, N\} \) contains a monochromatic arithmetic progression of length at least \( k \). Rabung gave a method to obtain lower bounds on \( W(2, k) \) based on quadratic residues, and performed computations on all primes no greater than \( 20117 \). By improving the efficiency of the algorithm of Rabung, we perform the computation for all primes up to \( 6 \times 10^7 \), and obtain lower bounds on \( W(2, k) \) for \( k \) between \( 11 \) and \( 23 \).