The complexity of P4-decomposition of regular graphs and multigraphs

Abstract

Let G denote a multigraph with edge set E(G), let μ(G) denote the maximum edge multiplicity in G, and let Pk denote the path on k vertices. Heinrich et al.(1999) showed that P4 decomposes a connected 4-regular graph G if and only if /E(G)/ is divisible by 3. We show that P4 decomposes a connected 4-regular multigraph G with μ(G) ≤ 2 if and only if no 3 vertices of G induce more than 4 edges and /E(G)/ is divisible by 3. Oksimets (2003) proved that for all integers k ≥ 3, P4 decomposes a connected 2k-regular graph G if and only if /E(G)/ is divisible by 3. We prove that for all integers k ≥ 2, the problem of determining if P4 decomposes a (2k + 1)-regular graph is NPComplete. El-Zanati et al.(2014) showed that for all integers k ≥ 1, every 6k-regular multigraph with μ (G) ≤ 2k has a P4-decomposition. We show that unless P = NP, this result is best possible with respect to μ (G) by proving that for all integers k ≥ 3 the problem of determining if P4 decomposes a 2k-regular multigraph with μ (G) ≤ [2k/3] + 1 is NP-Complete.