Related papers: Paths With Three Blocks In Digraphs
Let f(k) denote the maximum such that every simple undirected graph containing two vertices s,t and k edge-disjoint s-t paths, also contains two vertices u,v and f(k) independent u-v paths. Here, a set of paths is independent if none of…
A path system $\mathscr{P}$ in a graph $G=(V,E)$ is a collection of paths, with exactly one path between any two vertices in $V$. A path system is said to be consistent if it is closed under subpaths. We say that a path system $\mathscr{P}$…
Given a line arrangement $\cal A$ with $n$ lines, we show that there exists a path of length $n^2/3 - O(n)$ in the dual graph of $\cal A$ formed by its faces. This bound is tight up to lower order terms. For the bicolored version, we…
An interval coloring of a graph G is a proper coloring of E(G) by positive integers such that the colors on the edges incident to any vertex are consecutive. A (3,4)-biregular bigraph is a bipartite graph in which each vertex of one part…
An oriented cycle is an orientation of a undirected cycle. We first show that for any oriented cycle $C$, there are digraphs containing no subdivision of $C$ (as a subdigraph) and arbitrarily large chromatic number. In contrast, we show…
A linearly ordered (LO) $k$-colouring of an $r$-uniform hypergraph assigns an integer from $\{1, \ldots, k \}$ to every vertex so that, in every edge, the (multi)set of colours has a unique maximum. Equivalently, for $r=3$, if two vertices…
A planar graph can be embedded in a piecewise linear manifold, and the lattice on each linear piece can be colored with 3-coloring. If a planar graph can be colored with multiple 3-coloring, i.e. coloring the graph in pieces with different…
For integers $k\ge 2$ and $\ell\ge 0$, a $k$-uniform hypergraph is called a loose path of length $\ell$, and denoted by $P_\ell^{(k)}$, if it consists of $\ell $ edges $e_1,\dots,e_\ell$ such that $|e_i\cap e_j|=1$ if $|i-j|=1$ and $e_i\cap…
The paper shows the existence of a family of directed strongly regular graphs with parameters (22, 9, 6, 3, 4). The adjacency matrices of the found digraphs are composed of $3\times 3$ circulant blocks. The automorphism group of all the…
All planar graphs are 4-colorable and 5-choosable, while some planar graphs are not 4-choosable. Determining which properties guarantee that a planar graph can be colored using lists of size four has received significant attention. In terms…
A {\it path covering} of a graph $G$ is a set of vertex disjoint paths of $G$ containing all the vertices of $G$. The {\it path covering number} of $G$, denoted by $P(G)$, is the minimum number of paths in a path covering of $G$. An {\sl…
The proof uses the property that the vertices of a triangulated planar graph can be four coloured if the triangles can have a +1 or -1 orientation so that the sum of the triangle orientations around each vertex is a multiple of 3. Such…
The arc graph $\delta(G)$ of a digraph $G$ is the digraph with the set of arcs of $G$ as vertex-set, where the arcs of $\delta(G)$ join consecutive arcs of $G$. In 1981, Poljak and R\"{o}dl characterised the chromatic number of $\delta(G)$…
We introduce a class of pairs of graphs consisting of two cliques joined by an arbitrary number of edges. The members of a pair have the property that the clique-bridging edge-set of one graph is the complement of that of the other. We…
The dimension of a block design is the maximum positive integer $d$ such that any $d$ of its points are contained in a proper subdesign. Pairwise balanced designs PBD$(v,K)$ have dimension at least two as long as not all points are on the…
Let $H$ be a digraph possibly with loops, $D$ a digraph without loops, and $\rho : A(D) \rightarrow V(H)$ a coloring of $A(D)$ ($D$ is said to be an $H$-colored digraph). If $W=(x_{0}, \ldots , x_{n})$ is a walk in $D$, and $i \in \{ 0,…
New results on pentagonal geometries PENT(k,r) with block sizes k = 3 or k = 4 are given. In particular we completely determine the existence spectra for PENT(3,r) systems with the maximum number of opposite line pairs as well as those…
A $P_m$ path in a graph is a path on $m$ vertices. A $P_m$ system of order $n>1$ is a partition of the edges of the complete graph $K_n$ into $P_m$ paths. A $P_m$ system is said to be $k$-colourable if the vertex set of $K_n$ can be…
A directed path whose edges are assigned labels "up", "down", "right", or "left" is called \emph{four-directional}, and \emph{three-directional} if at most three out of the four labels are used. A \emph{direction-consistent embedding} of an…
We study the structure of the space $\Omega_3(G)$ of $\partial$-invariant 3-paths in a directed graph $G$. We prove that $\Omega_3(G)$ admits a basis consisting of trapezohedral paths $\tau_m$ ($m \ge 2$) and their merging images. Moreover,…