Even-cycle decompositions of graphs with no odd-K4K_4-minor

An even-cycle decomposition of a graph G is a partition of E(G) into cycles of even length. Evidently, every Eulerian bipartite graph has an even-cycle decomposition. Seymour (1981) proved that every 2-connected loopless Eulerian planar graph with an even number of edges also admits an even-cycle decomposition. Later, Zhang (1994) generalized this to graphs with no $K_5$-minor. Our main theorem gives sufficient conditions for the existence of even-cycle decompositions of graphs in the absence of odd minors. Namely, we prove that every 2-connected loopless Eulerian odd-$K_4$-minor-free graph with an even number of edges has an even-cycle decomposition. This is best possible in the sense that `odd-$K_4$-minor-free' cannot be replaced with `odd-$K_5$-minor-free.' The main technical ingredient is a structural characterization of the class of odd-$K_4$-minor-free graphs, which is due to Lov\'asz, Seymour, Schrijver, and Truemper.Comment: 17 pages, 6 figures; minor revisio

Triangulations of the torus with at most two odd vertices: structure and coloring

This thesis consists of two parts. In the first part, we give a simple geometric description of the set $mathcal G(5,5,8)$ of toroidal triangulations, all of whose vertices have degree six, except for two of degree five and one of degree eight. The motivation for studying such family is provided by Gr"unbaum coloring application described below. Each such triangulation is described by a cut-and-glue construction starting from an infinite triangular grid. In particular, we show that the members of $mathcal G(5,5,8)$ are obtained from a toroidal 6-regular graph (three parameters) by cutting out a special disk, described with two parameters, and ``stitching" along the cut. To achieve that, we develop some techniques and define some invariants to study the cycles of toroidal triangulations. We also introduce some special triangulated disks called ``blocks" and show how to detect their presence in a triangulation. Then, we show the existence of a special path that identifies the ``cut". Also, the graphs in $mathcal G(5,5,8)$ are classified into several families, based on the existence of some special cycles, containing a vertex of degree eight. Each family is, further, described by a schema for gluing together a few blocks. The second part regards coloring. A Gr"unbaum coloring of a graph $G$ which triangulates a surface is a 3-edge-coloring of $G$ in which every face is incident to three edges of different colors. In 1968 Gr"unbaum conjectured a generalization of the Four Color Theorem: every simple triangulation of every orientable surface has a Gr"unbaum coloring. In 2008 Kochol discovered counterexamples to Gr"unbaum\u27s conjecture on every orientable surface of genus at least five. Gr"unbaum\u27s conjecture is still believed to be true for the torus. We verify ``weak" Gr"unbaum conjecture for three families of triangulations in higher surfaces that, to our knowledge, are the only known families of triangulations with unbounded facewidth that are not 4-colorable. Also, as an application of our description, we propose a method by which to verify the Gr"unbaum\u27s conjecture for $mathcal G(5,5,8)$

Strongly Even-Cycle Decomposable Graphs

A graph is strongly even-cycle decomposable if the edge set of every subdivision with an even number of edges can be partitioned into cycles of even length. We prove that several fundamental composition operations that preserve the property of being Eulerian also yield strongly even-cycle decomposable graphs. As an easy application of our theorems, we give an exact characterization of the set of strongly even-cycle decomposable cographs.SCOPUS: ar.jFLWINinfo:eu-repo/semantics/publishe

Silver Cubes

An n × n matrix A is said to be silver if, for i = 1,2,...,n, each symbol in {1,2,...,2n − 1} appears either in the ith row or the ith column of A. The 38th International Mathematical Olympiad asked whether a silver matrix exists with n = 1997. More generally, a silver cube is a triple (K d n,I,c) where I is a maximum independent set in a Cartesian power of the complete graph Kn, and c: V (K d n) → {1,2,...,d(n − 1) + 1} is a vertex colouring where, for v ∈ I, the closed neighbourhood N[v] sees every colour. Silver cubes are related to codes, dominating sets, and those with n a prime power are also related to finite geometry. We present here algebraic constructions, small examples, and a product construction. The nonexistence of silver cubes for d = 2 and some values of n, is proved using bounds from coding theory