Graphs with cyclomatic number three having panconnected square

The square of a graph [Formula: see text] is the graph obtained from [Formula: see text] by adding edges joining those pairs of vertices whose distance from each other in [Formula: see text] is two. If [Formula: see text] is connected, then the cyclomatic number of [Formula: see text] is defined as [Formula: see text]. Graphs with cyclomatic number not more than [Formula: see text] whose square are panconnected have been characterized, among other things, in [G. L. Chia, S. H. Ong and L. Y. Tan, On graphs whose square have strong Hamiltonian properties, Discrete Math. 309 (2009) 4608–4613, G. L. Chia, W. Hemakul and S. Singhun, Graphs with cyclomatic number two having panconnected square, Discrete Math. 311 (2011) 850–855]. Here, we show that if [Formula: see text] has cyclomatic number [Formula: see text] and [Formula: see text] is panconnected, then [Formula: see text] is one of the eight families of graphs, [Formula: see text], defined in the paper. Further, we obtain necessary and sufficient conditions for three larger families of graphs (which contains [Formula: see text] as special cases) whose square are panconnected.

Hamiltonian Cycles in the Square of a Graph

We show that under certain conditions the square of the graph obtained by identifying a vertex in two graphs with hamiltonian square is also hamiltonian. Using this result, we prove necessary and sufficient conditions for hamiltonicity of the square of a connected graph such that every vertex of degree at least three in a block graph corresponds to a cut vertex and any two these vertices are at distance at least four.

The Chromatic Number of Graph Powers

It is shown that the maximum possible chromatic number of the square of a graph with maximum degree d and girth g is (1 +o(1))d2 if g = 3, 4, 5 or 6, and is Θ(d2 / log d) if g [ges ] 7. Extensions to higher powers are considered as well.