In 1857 Arthur Cayley (1821–1895) published a paper [9] that introduces the term “tree” to describe the logical branching that occurs when iterating the fundamental process of (partial) differentiation. When discussing the composition of four symbols that involve derivatives, Cayley writes “But without a more convenient notation, it would be difficult to find [their] corresponding expressions ....

We survey results in geometric network design theory, including algorithms for constructing minimum spanning trees and low-dilation graphs.

Two spanning trees rooted at some vertex r in a graph G are said to be independent if for each vertex v of G, v ≠ r, the paths from r to v in two trees are vertex-disjoint. A set of spanning trees of G is said to be independent if they are pairwise independent. A set of independent spanning trees is optimal if the average path length of the trees is the minimum. Any k-dimensional hypercube has ...

Given a graph G, a designated vertex r and a natural number k, we wish to find k “independent” spanning trees ofG rooted at r, that is, k spanning trees such that, for any vertex v, the k paths connecting r and v in the k trees are internally disjoint in G. In this paper we give a linear-time algorithm to find k independent spanning trees in a k-connected maximal planar graph rooted at any desi...

Coursebooks discussing graph algorithms usually have a chapter on minimum spanning trees. It usually contains Prim’s and Kruskal’s algorithms [1, 2] but often lacks other applications. This type of problem is rarely present at informatics competitions or in tests in secondary or higher level informatics education This article is aimed at describing some competition tasks that help us prove that...

S of the Ghent Graph Theory Workshop on Longest Paths and Longest Cycles Kathie Cameron Degree-constrained spanning trees 2 Jan Goedgebeur Finding minimal obstructions to graph coloring 3 Jochen Harant On longest cycles in essentially 4-connected planar graphs 3 Frantǐsek Kardoš Barnette was right: not only fullerene graphs are Hamiltonian 4 Gyula Y. Katona Complexity questions for minimally t-...

Let Zm be the cyclic group of order m ≥ 3. A graph G is Zm-connected if G has an orientation D such that for any mapping b : V (G) 7→ Zm with ∑ v∈V (G) b(v) = 0, there exists a mapping f : E(G) 7→ Zm − {0} satisfying ∑ e∈E+ D (v) f(e) − ∑ e∈E− D (v) f(e) = b(v) in Zm for any v ∈ V (G); and a graph G is strongly Zm-connected if, for any mapping θ : V (G) → Zm with ∑ v∈V (G) θ(v) = |E(G)| in Zm, ...