The partial list colouring conjecture due to Albertson, Grossman, and Haas [1] states that for every s-choosable graph G and every assignment of lists of size t, 1 6 t 6 s, to the vertices of G there is an induced subgraph of G on at least t|V (G)| s vertices which can be properly coloured from these lists. In this paper, we show that the partial list colouring conjecture holds true for certain...

The distinguishing number D(G) of a graph is the least integer d such that there is a d-labeling of the vertices of G that is not preserved by any nontrivial automorphism of G. We show that the distinguishing number of the square and higher powers of a connected graph G 6= K2,K3 with respect to the Cartesian product is 2. This result strengthens results of Albertson [1] on powers of prime graph...

1 The distinguishing number D(G) of a graph G is the least integer d such that there is a d-labeling 2 of the vertices of G which is not preserved by any nontrivial automorphism. For a graph G let Gr 3 be the r th power of G with respect to the Cartesian product. It is proved that D(Gr ) = 2 for any 4 connected graph G with at least 3 vertices and for any r ≥ 3. This confirms and strengthens a ...

Measures of the irregularity of chemical graphs could be helpful for QSAR/QSPR studies and for the descriptive purposes of biological and chemical properties, such as melting and boiling points, toxicity and resistance. Here we consider the following four established irregularity measures: the irregularity index by Albertson, the total irregularity, the variance of vertex degrees and the Collat...

Let G be a graph embedded on a surface with all noncontractible cycles sufficiently long, called a locally planar graph. With Michael Albertson we have shown that if a set of vertices of G, at sufficiently large, mutual distance, are precolored with five colors, then this coloring extends to a 5-coloring of G. The precolored extension 5-color theorem holds if bipartite subgraphs, sufficiently f...

Chemical graph theory is currently expanding the use of topological indices to numerically encode chemical structure. The prediction characteristics provided by structure molecule a key feature these indices. concepts from are presented in brief discussion one its many applications chemistry, namely, quantitative structure-activity relationship (QSAR) studies and structure-property (QSPR) studi...