Unbalanced subtrees in binary rooted ordered and un-ordered trees

Binary rooted trees, both in the ordered and in the un-ordered case, are well studied structures in the field of combinatorics. The aim of this work is to study particular patterns in these classes of trees. We consider completely unbalanced subtrees, where unbalancing is measured according to the so-called Colless's index. The size of the biggest unbalanced subtree becomes then a new parameter with respect to which we find several enumerations

On the excessive [m]-index of a tree

The excessive [m]-index of a graph G is the minimum number of matchings of size m needed to cover the edge-set of G. We call a graph G [m]-coverable if its excessive [m]-index is finite. Obviously the excessive [1]-index is |E(G)| for all graphs and it is an easy task the computation of the excessive [2]-index for a [2]-coverable graph. The case m=3 is completely solved by Cariolaro and Fu in 2009. In this paper we prove a general formula to compute the excessive [4]-index of a tree and we conjecture a possible generalization for any value of m. Furthermore, we prove that such a formula does not work for the excessive [4]-index of an arbitrary graph.Comment: 12 pages, 7 figures, to appear in Discrete Applied Mathematic

Bandwidth and density for block graphs

The bandwidth of a graph G is the minimum of the maximum difference between adjacent labels when the vertices have distinct integer labels. We provide a polynomial algorithm to produce an optimal bandwidth labeling for graphs in a special class of block graphs (graphs in which every block is a clique), namely those where deleting the vertices of degree one produces a path of cliques. The result is best possible in various ways. Furthermore, for two classes of graphs that are ``almost'' caterpillars, the bandwidth problem is NP-complete.Comment: 14 pages, 9 included figures. Note: figures did not appear in original upload; resubmission corrects thi

\u3ci\u3eOwlet Caterpillars of Eastern North America.\u3c/i\u3e David L. Wagner, Dale F. Schweitzer, J Bolling Sullivan & Richard C. Reardon. 2011. Princeton University Press, 576 pp., soft cover, 8 by 10.

(excerpt) You may be wondering what an owlet caterpillar is, since “owlet” is not mentioned in some books about insects. It is a general name for moths in the family Noctuidae and is nicely defined by Marshall (2006) as: “nocturnal moths are sometimes called owlet moths (noctua means owl in Latin) because of the way their eyes pick up and reflect the smallest amount of light, shining brightly in contrast with the usually inconspicuous body and forewings”