Degree lower bounds for a graph ordering principle∗

We prove that a family of polynomials encoding a Graph Ordering Principle (GOP(G)) requires refutations in Polynomial Calculus (PC) and Polynomial Calculus with Resolution (PCR) of degree Ω(n). Here n is the number of vertices in the graph G. This is the first linear degree lower bound for PC and PCR refutations for ordering principles. Our result answers to a question raised by Bonet and Galesi in [9, 10] and implies that the size-degree tradeoff for PCR of Alekhnovich et al. [7, 3] is optimal, since there are polynomial size PCR refutatations for GOP(G). We then introduce an algebraic proof system PCRk which combines together Polynomial Calculus (PC) and k-DNF Resolution (RESk). As a corollary of the previous lower bound and using techniques developed for RESk, we show a size hierarchy theorem for PCRk: PCRk is exponentially separated from PCRk+1. Finally we show that in PCRk random formulas in conjunctive normal form (3-CNF) are hard to refute.