On the graceful labelling of triangular cacti conjecture

A graceful labelling of a graph with n edges is a vertex labelling where the induced set of edge weights is {1, . . . , n}. A near graceful labelling is almost the same, the difference being that the edge weights are {1, 2, . . . , n − 1, n + 1}. In both cases, the weight of an edge is the absolute difference between its two vertex labels. Rosa [8] in 1988 conjectured that all triangular cacti are either graceful or near graceful. He also suggested the use of Skolem sequences to label some types of triangular cacti. In this paper, we verify the conjecture for two families of triangular cacti, and extend the discussion for further research. Particular constructions of Skolem sequences are discussed, as well as a technique using Langford sequences to obtain Skolem and hooked Skolem sequences with specific sub-sequences. These special sequences are used to gracefully label the two families which are the focus of the paper.