Tight Bounds for Cycle-Edge Decompositions and Covers

An old conjecture of Erd{\H{o}}s and Gallai states that every $n$ vertex graph can be decomposed, that is $E(G)$ can be partitioned, into $O(n)$ cycles and edges. The covering version of this conjecture was proven by Pyber in 1985, where it was shown that all graphs can be covered by $n-1$ cycles and edges. The best upper bound on the number of cycles and edges required to decompose any graph is $O(n\log^*(n))$, which was recently shown by Buci{\'c} and Montgomery in 2023. Here $\log^*(n)$ denotes the iterated logarithm function. Meanwhile, a construction of Erd\H{o}s demonstrate that there exists graphs which require $(\frac{3}{2}-o(1))n$ cycles and edges to be decomposed. We prove all graphs with maximum degree at most $4$ can be decomposed into $n-1$ or fewer cycles and edges. We also show that every $n$ vertex claw-free graph can be decomposed into $n-1$ or fewer $2$-regular subgraphs and edges. Finally, we prove that every graph $G$ containing a cycle can be covered by $n-2$ or fewer cycles and edges. This improves Pyber's covering theorem by proving that $n-1$ cycles and edges are required only for trees.