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Let G denote a multigraph with edge set E(G), let &amp;micro;(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 &amp;micro;(G) &amp;#x2264;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 &amp;#x2265;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 &amp;#x2265;2, the problem of determining if P4 decomposes a (2k + 1)-regular graph is NP-Complete. El-Zanati et al.(2014) showed that for all integers k &amp;#x2265;1, every 6k-regular multigraph with &amp;micro;(G) &amp;#x2264;2k has a P4-decomposition. We show that unless P = NP, this result is best possible with respect to &amp;micro;(G) by proving that for all integers k &amp;#x2265;3 the problem of determining if P4 decomposes a 2k-regular multigraph with &amp;micro;(G) &amp;#x2264;&amp;#x230A;2k&amp;#8202;/&amp;#8202;3 &amp;#x230B;+ 1 is NP-Complete.

The complexity of $P$<sub>4</sub>-decomposition of regular graphs and multigraphs