New Advances in Kotzig's Conjecture

Abstract

In 1973 Kotzig conjectures that the complete graph $K_{2n+1}$ can be cyclically decomposed into $2n+1$ copies of any tree of size $n$. Rosa proved that this decomposition exists if and only if there exists a $\rho$-labeling of the tree. In this work we prove that if $T'$ is a graceful tree, then any tree $T$ obtained from $T'$ by attaching a total of $k \geq 1$ pendant vertices to any collection of $r$ vertices of $T'$, where $1 \leq r \leq k$, admits a $\rho$-labeling. As a consequence of this result, many new families of trees with this kind of labeling are produced, which indicates the strong potential of this result. Moreover, the technique used to prove this result, gives us an indication of how to determine whether a given tree of size $n$ decomposes the complete graph $K_{2n+1}$. We also prove the existence of a $\rho$-labeling for two subfamilies of lobsters and present a method to produce $\rho$-labeled trees attaching pendant vertices and pendant copies of the path $P_3$ to some of the vertices of any graceful tree.\\ In addition, for any given tree $T$, we use bipartite labelings to show that this tree is a spanning tree of a graph $G$ that admits an $\alpha$-labeling. This is not a new result; however, the construction presented here optimizes (reduces) the size of $G$ with respect to all the similar results that we found in the literature.