Related papers: Lattice Path Matroids are 3-Colorable

We count the number of lattice paths lying under a cyclically shifting piecewise linear boundary of varying slope. Our main result extends well known enumerative formulae concerning lattice paths, and its derivation involves a classical…

Let $c$ be an edge-colouring of a graph $G$ such that for every vertex $v$ there are at least $d \ge 2$ different colours on edges incident to $v$. We prove that $G$ contains a properly coloured path of length 2d or a properly coloured…

A matroid M is cyclically orderable if there is a cyclic permutation of the elements of M such that any r consecutive elements form a basis in M. An old conjecture of Kajitani, Miyano, and Ueno states that a matroid M is cyclically…

We show that the minimum number of orientations of the edges of the n-vertex complete graph having the property that every triangle is made cyclic in at least one of them is $\lceil\log_2(n-1)\rceil$. More generally, we also determine the…

In this note we introduce a family of polynomials on a matroid derived from chain Tutte polynomials which generalize the classic and ubiquitous characteristic polynomial. We show that the coefficients of these polynomials alternate and…

We consider the problem of counting the number of 3-colourings of the edges (bonds) of the 4-8 lattice and the 3-12 lattice. These lattices are Archimedean with coordination number 3, and can be regarded as decorated versions of the square…

A vertex colouring of a graph $G$ is "nonrepetitive" if $G$ contains no path for which the first half of the path is assigned the same sequence of colours as the second half. Thue's famous theorem says that every path is nonrepetitively…

For all positive integers $t$ exceeding one, a matroid has the cyclic $(t-1,t)$-property if its ground set has a cyclic ordering $\sigma$ such that every set of $t-1$ consecutive elements in $\sigma$ is contained in a $t$-element circuit…

Halin proved that every minimally $k$-connected graph has a vertex of degree $k$. More generally, does every minimally vertically $k$-connected matroid have a $k$-element cocircuit? Results of Murty and Wong give an affirmative answer when…

A famous theorem of Galvin states that the list chromatic number of the intersection of two partition matroids equals its chromatic number. Kiraly and Berczi et. al. conjectured that this equality holds for any two matroids. We prove this…

In 1994, Thomassen proved that every planar graph is 5-list-colorable. In 1995, Thomassen proved that every planar graph of girth at least five is 3-list-colorable. His proofs naturally lead to quadratic-time algorithms to find such…

Knot colorings are one of the simplest ways to distinguish knots, dating back to Reidemeister, and popularized by Fox. In this mostly expository article, we discuss knot invariants like colorability, knot determinant and number of…

While every plane triangulation is colourable with three or four colours, Heawood showed that a plane triangulation is 3-colourable if and only if every vertex has even degree. In $d \geq 3$ dimensions, however, every $k \geq d+1$ may occur…

We prove that for all nonnegative integers k,s there exists c with the following property. Let G be a graph with clique number at most k and chromatic number more than c. Then for every vertex-colouring (not necessarily optimal) of G, some…

A rational Dyck path of type $(m,d)$ is an increasing unit-step lattice path from $(0,0)$ to $(m,d) \in \mathbb{Z}^2$ that never goes above the diagonal line $y = (d/m)x$. On the other hand, a positroid of rank $d$ on the ground set $[d+m]$…

An integral lattice which is generated by some vectors of norm $q$ is called $q$-lattice. Classification of 3-lattices of dimension at most four is given by Mimura (On 3-lattice, 2006). As a expansion, we give a classification of 3-lattices…

We consider posets of lattice paths (endowed with a natural order) and begin the study of such structures. We give an algebraic condition to recognize which ones of these posets are lattices. Next we study the class of Dyck lattices (i.e.,…

Sopena in his survey [E. Sopena, The oriented chromatic number of graphs: A short survey, preprint 2013] writes, without any proof, that an oriented cycle $\vec C$ can be colored with three colors if and only if $\lambda(\vec C)=0$, where…

Let $K_{n}^{c}$ denote a complete graph on $n$ vertices whose edges are colored in an arbitrary way. Let $\Delta^{\mathrm{mon}} (K_{n}^{c})$ denote the maximum number of edges of the same color incident with a vertex of $K_{n}^{c}$. A…

Consider lattice paths in Z^2 taking unit steps north (N) and east (E). Fix positive integers r,s and put an equivalence relation on points of Z^2 by letting v,w be equivalent if v - w = m (r,s) for some m in Z. Call a lattice path valid if…