Tutte's 5-Flow Conjecture for Highly Cyclically Connected Cubic Graphs

In 1954, Tutte conjectured that every bridgeless graph has a nowhere-zero 5-flow. Let $\omega$ be the minimum number of odd cycles in a 2-factor of a bridgeless cubic graph. Tutte's conjecture is equivalent to its restriction to cubic graphs with $\omega \geq 2$. We show that if a cubic graph $G$ has no edge cut with fewer than ${5/2} \omega - 1$ edges that separates two odd cycles of a minimum 2-factor of $G$, then $G$ has a nowhere-zero 5-flow. This implies that if a cubic graph $G$ is cyclically $n$-edge connected and $n \geq {5/2} \omega - 1$, then $G$ has a nowhere-zero 5-flow

Stable dominating circuits in snarks

AbstractSnarks are cyclically 4-edge-connected cubic graphs with girth at least 5 and with no 3-edge-coloring. We construct snarks with a (dominating) circuit C so that no other circuit C′ satisfies V(C)⊆V(C′). These graphs are of interest because two known conjectures about graphs can be reduced on them. The first one is Sabidusi's Compatibility Conjecture which suggests that given an eulerian trail T in an eulerian graph G without 2-valent vertices, there exists a decomposition of G into circuits such that consecutive edges in T belong to different circuits. The second conjecture is the Fixed-Circuit Cycle Double-Cover Conjecture suggesting that every bridgeless graph has a cycle double cover which includes a fixed circuit

Bounds of characteristic polynomials of regular matroids

A regular chain group $N$ is the set of integral vectors orthogonal to rows of a matrix representing a regular matroid, i.e., a totally unimodular matrix. Introducing canonical forms of an equivalence relation generated by $N$ and a special basis of $N$, we improve several results about polynomials counting elements of $N$ and find new bounds and formulas for these polynomials

Compatible systems of representatives

AbstractThe main result of this paper can be quickly described as follows. Let G be a bipartite graph and assume that for any vertex v of G a strongly base orderable matroid is given on the set of edges adjacent with v. Call a subgraph of G a system of representatives of G if the edge neighborhood of each vertex of this subgraph is independent in the corresponding matroid. Two systems of representatives we call compatible if they have no common edge. We give a necessary and sufficient condition for G to have k pairwise compatible systems of representatives with at least d edges. Unfortunately, this condition is not sufficient if we deal with arbitrary matroids. Furthermore, we establish a listing variant of the Edmonds' covering theorem for strongly base orderable matroids

A cyclically 6-edge-connected snark of order 118

AbstractWe present a cyclically 6-edge-connected snark of order 118, thereby illustrating a new method of constructing snarks

Measures of edge-uncolorability

The resistance $r(G)$ of a graph $G$ is the minimum number of edges that have to be removed from $G$ to obtain a graph which is $\Delta(G)$-edge-colorable. The paper relates the resistance to other parameters that measure how far is a graph from being $\Delta$-edge-colorable. The first part considers regular graphs and the relation of the resistance to structural properties in terms of 2-factors. The second part studies general (multi-) graphs $G$. Let $r_v(G)$ be the minimum number of vertices that have to be removed from $G$ to obtain a class 1 graph. We show that $\frac{r(G)}{r_v(G)} \leq \lfloor \frac{\Delta(G)}{2} \rfloor$, and that this bound is best possible.Comment: 9 page

A note about the dominating circuit conjecture

AbstractThe dominating circuit conjecture states that every cyclically 4-edge-connected cubic graph has a dominating circuit. We show that this is equivalent to the statement that any two edges of such a cyclically 4-edge-connected graph are contained in a dominating circuit

Nonextendible Latin Cuboids

We show that for all integers m >= 4 there exists a 2m x 2m x m latin cuboid that cannot be completed to a 2mx2mx2m latin cube. We also show that for all even m > 2 there exists a (2m-1) x (2m-1) x (m-1) latin cuboid that cannot be extended to any (2m-1) x (2m-1) x m latin cuboid

A new Lenstra-type Algorithm for Quasiconvex Polynomial Integer Minimization with Complexity 2^O(n log n)

We study the integer minimization of a quasiconvex polynomial with quasiconvex polynomial constraints. We propose a new algorithm that is an improvement upon the best known algorithm due to Heinz (Journal of Complexity, 2005). This improvement is achieved by applying a new modern Lenstra-type algorithm, finding optimal ellipsoid roundings, and considering sparse encodings of polynomials. For the bounded case, our algorithm attains a time-complexity of s (r l M d)^{O(1)} 2^{2n log_2(n) + O(n)} when M is a bound on the number of monomials in each polynomial and r is the binary encoding length of a bound on the feasible region. In the general case, s l^{O(1)} d^{O(n)} 2^{2n log_2(n) +O(n)}. In each we assume d>= 2 is a bound on the total degree of the polynomials and l bounds the maximum binary encoding size of the input.Comment: 28 pages, 10 figure

Covering planar graphs with forests

AbstractWe study the problem of covering graphs with trees and a graph of bounded maximum degree. By a classical theorem of Nash-Williams, every planar graph can be covered by three trees. We show that every planar graph can be covered by two trees and a forest, and the maximum degree of the forest is at most 8. Stronger results are obtained for some special classes of planar graphs