Packing closed trails into dense graphs. (Q1405105)

Let \(G_{1}\) and \(G_{2}\) be graphs. A mapping \(f:V(G_{1})\rightarrow V(G_{2})\) (not necessary injective) is called packing if it induces an injection from the set \(E(G_{1})\) to \(E(G_{2})\). This means that \((x,y)\) is an edge of \(G_{1}\) if and only if \((f(x),f(y))\) is an edge of \(G_{2}\). From this definition it is clear that a packing of a cycle is a closed trail. The main result of the paper is the existence of a packing of cycles with given lengths to dense graphs. More precisely, there exists an integer \(N\) and number \(\varepsilon >0\) such that for any graph \(G\) with \(n\) vertices \( (n\geq N)\) and all vertices of even degree at least \((1-\varepsilon )n\) it is possible to pack into \(G\) any disjoint union of cycles \(C_{1}\cup C_{2}\cup \dots\cup C_{t}\) where \(\sum\limits_{i=1}^{t}m_{i}=\left| E(G)\right| \) and \(m_{i}\) is the length of the cycle \(C_{i}\) for \(1\leq i\leq t\). A similar result for graphs \(K_{n}\) with\ odd \(\;n\) was proved by the author in an earlier paper. This result is closely related to Alspach's hypothesis where instead of packing a decomposition into disjoint cycles with given lengths is needed.