We prove that there is a Steiner triple system T such that every simple cubic graph can have its edges coloured by points of T in such a way that for each vertex the colours of the three incident edges form a triple in T . This result complements the result of Holroyd and Škoviera that every bridgeless cubic graph admits a similar colouring by any Steiner triple system of order greater than 3. ...

An S-colouring of a cubic graph G is an edge-colouring of G by points of a Steiner triple system S such that the colours of any three edges meeting at a vertex form a block of S. In this note we present an infinite family of point-intransitive Steiner triple systems S such that (1) every simple cubic graph is S-colourable and (2) no proper subsystem of S has the same property. Only one point-in...

How to measure the complexity of a finite set of vectors embedded in a multidimensional space? This is a non-trivial question which can be approached in many different ways. Here we suggest a set of data complexity measures using universal approximators, principal cubic complexes. Principal cubic complexes generalise the notion of principal manifolds for datasets with nontrivial topologies. The...

We study a new problem for cubic graphs: bipartization of a cubic graph Q by deleting sufficiently large independent set I. It can be expressed as follows: Given a connected n-vertex tripartite cubic graph Q = (V,E) with independence number α(Q), does Q contain an independent set I of size k such that Q− I is bipartite? We are interested for which value of k the answer to this question is affir...

In this paper we have shown without assuming the four color theorem of planar graphs that every (bridgeless) cubic planar graph has a threeedge-coloring. This is an old-conjecture due to Tait in the squeal of efforts in settling the four-color conjecture at the end of the 19th century. We have also shown the applicability of our method to another well-known three edge-coloring conjecture on cub...

The cutoff phenomenon describes a sharp transition in the convergence of an ergodic finite Markov chain to equilibrium. Of particular interest is understanding this convergence for the simple random walk on a bounded-degree expander graph. The first example of a family of bounded-degree graphs where the random walk exhibits cutoff in total-variation was provided only very recently, when the aut...

There are many hard conjectures in graph theory, like Tutte’s 5-flow conjecture, and the 5-cycle double cover conjecture, which would be true in general if they would be true for cubic graphs. Since most of them are trivially true for 3-edge-colorable cubic graphs, cubic graphs which are not 3-edge-colorable, often called snarks, play a key role in this context. Here, we survey parameters measu...

We prove that for graphs of order n, minimum degree δ ≥ 2 and girth g ≥ 5 the domination number γ satisfies γ ≤ ( 1 3 + 2 3g ) n. As a corollary this implies that for cubic graphs of order n and girth g ≥ 5 the domination number γ satisfies γ ≤ ( 44 135 + 82 135g ) n which improves recent results due to Kostochka and Stodolsky (An upper bound on the domination number of n-vertex connected cubic...

A cubic graph G is S-edge-colorable for a Steiner triple system S if its edges can be colored with points of S in such a way that the points assigned to three edges Electronic Notes in Discrete Mathematics 29 (2007) 23–27 1571-0653/$ – see front matter © 2007 Elsevier B.V. All rights reserved. www.elsevier.com/locate/endm doi:10.1016/j.endm.2007.07.005 sharing a vertex form a triple in S. We sh...