Weighted maximum matchings and optimal equi-difference conflict-avoiding codes

A conflict-avoiding code (CAC) C of length n and weight k is a collection of k-subsets of Zn such that Δ(x) ∩ Δ(y) = ∅ for any x, y ∈ C and x = y, where Δ(x) = {a − b : a, b ∈ x, a = b}. Let CAC(n, k) denote the class of all CACs of length n and weight k. A CAC C ∈ CAC(n, k) is said to be equi-difference if any codeword x ∈ C has the form {0, i, 2i, . . . , (k − 1)i}. A CAC with maximum size is called optimal. In this paper we propose a graphical characterization of an equi-difference CAC, and then provide an infinite number of optimal equi-difference CACs for weight four.