On the structure and deficiency of k-trees with bounded degree

A?proper edge colouring of a?graph with natural numbers is consecutive if colours of edges incident with each vertex form a?consecutive interval of integers. The?deficiency d e f ( G ) of a?graph G is the?minimum number of pendant edges whose attachment to G makes it consecutively colourable. Since all 1-trees are consecutively colourable, in this paper we study the?deficiency of k -trees for k ? 2 . Our investigation establishes the?values of the?deficiency of all k -trees that have maximum degree bounded from above by 2 k , with k ? { 2 , 3 , 4 } . To obtain these results we consider the?structure of k -trees with bounded degree and the?deficiency of general graphs of odd order. In the?first case we show that for n ? 2 k + 3 the?structure of an? n -vertex k -tree with maximum degree not greater than 2 k is unique. In the second one we prove that for each n -vertex graph G of odd order the?inequality d e f ( G ) ? 1 2 ( | E ( G ) | - ( n - 1 ) Δ ( G ) ) holds. Both last mentioned results seem to be interesting in their own right.