G. Sethuraman

An antimagic labeling of a graph G with q edges is an injective mapping $f:E(G)\to \{1,2,\dots ,q\}$ such that the induced vertex label for each vertex is different, where the induced vertex label of a vertex u is ${\varphi }_f(u)=\sum _{e\in E(u)}f(e).$ Here, E(u) is the set of edges incident to the vertex u. In 1990, Hartsfield and Ringel conjectured that all trees except K₂ are antimagic. Still this conjecture is open. In this article, we prove that two recursive classes of trees called binomial tree B_k, $k\ge 2$ and Fibonacci tree F_h, $h\ge 2$ are antimagic. This result supports Hartsfield and Ringel conjecture.

The binomial tree B₀ $B_0$ <!--mathContainer--> <!--Loading Mathjax--> consists of a single vertex. The binomial tree B_k $B_k$ <!--mathContainer--> <!--Loading Mathjax--> is an ordered tree defined recursively. The binomial tree B_k $B_k$ <!--mathContainer--> <!--Loading Mathjax--> consists of two binomial trees B_k−1 $B_{k-1}$ <!--mathContainer--> <!--Loading Mathjax--> that are linked together: the root of one is the leftmost child of the root of the other. The popular Graceful Tree Conjecture states that every tree is graceful. In this paper, we show that binomial trees B_k $B_k$ <!--mathContainer--> <!--Loading Mathjax--> is graceful for every k≥0 $k\ge 0$ <!--mathContainer--> <!--Loading Mathjax--> .

Acharya (1982) proved that every connected graph can be embedded in a graceful graph. The generalization of this result that, any set of graphs can be packed into a graceful graph was proved by Sethuraman and Elumalai (2005). Recently, Sethuraman et al. (2016) have shown that, every tree can be embedded in an graceful tree. Inspired by these fundamental structural properties of graceful graphs, in this paper, we prove that any acyclic graph can be embedded in an unicyclic graceful graph. This result is proved algorithmically by constructing graceful unicyclic graphs from a given acyclic graph. Our result strongly supports the Truszczynski’s Conjecture that “All unicyclic graphs except the cycle $C_n$ with $n\equiv 1$ or 2(mod 4) are graceful”.