Related papers: Forbidden Patterns in Mixed Linear Layouts
Let $ex(n, P)$ be the maximum possible number of ones in any 0-1 matrix of dimensions $n \times n$ that avoids $P$. Matrix $P$ is called minimally non-linear if $ex(n, P) = \omega(n)$ but $ex(n, P') = O(n)$ for every strict subpattern $P'$…
A non-aligned drawing of a graph is a drawing where no two vertices are in the same row or column. Auber et al. showed that not all planar graphs have non-aligned drawings that are straight-line, planar, and in the minimal-possible $n\times…
A graph is circle if its vertices are in correspondence with a family of chords in a circle in such a way that every two distinct vertices are adjacent if and only if the corresponding chords have nonempty intersection. Even though there…
A graph is called an $(r,k)$-graph if its vertex set can be partitioned into $r$ parts, each having at most $k$ vertices and there is at least one edge between any two parts. Let $f(r,H)$ be the minimum $k$ for which there exists an…
A forbidden transition graph is a graph defined together with a set of permitted transitions i.e. unordered pair of adjacent edges that one may use consecutively in a walk in the graph. In this paper, we look for the smallest set of…
An edge-ordered graph is a graph with a linear ordering of its edges. Two edge-ordered graphs are equivalent if their is an isomorphism between them preserving the ordering of the edges. The edge-ordered Ramsey number $r_{edge}(H; q)$ of an…
The \emph{thinness} of a graph is a width parameter that generalizes some properties of interval graphs, which are exactly the graphs of thinness one. Graphs with thinness at most two include, for example, bipartite convex graphs. Many…
The emerging theory of graph limits exhibits an analytic perspective on graphs, showing that many important concepts and tools in graph theory and its applications can be described more naturally (and sometimes proved more easily) in…
Motivated by the definition of linear coloring on simplicial complexes, recently introduced in the context of algebraic topology \cite{Civan}, and the framework through which it was studied, we introduce the linear coloring on graphs. We…
A \emph{mixed interval graph} is an interval graph that has, for every pair of intersecting intervals, either an arc (directed arbitrarily) or an (undirected) edge. We are particularly interested in scenarios where edges and arcs are…
A signed graph is one that features two types of edges: positive and negative. Balanced signed graphs are those in which all cycles contain an even number of positive edges. In the adjacency matrix of a signed graph, entries can be $0$,…
A matching in a graph is uniquely restricted if no other matching covers exactly the same set of vertices. We establish tight lower bounds on the maximum size of a uniquely restricted matching in terms of order, size, and maximum degree.
A [z,r;g]-mixed cage is a mixed graph of minimum order such that each vertex has z in-arcs, z out-arcs, r edges, and it has girth g. We present an infinite family of mixed graphs with girth 6. This construction also provides an upper bound…
A vertex k-ranking is a labeling of the vertices of a graph with integers from 1 to k so any path connecting two vertices with the same label will pass through a vertex with a greater label. The rank number of a graph is defined to be the…
Given an edge-coloring of a graph, the palette of a vertex is defined as the set of colors of the edges which are incident with it. We define the palette index of a graph as the minimum number of distinct palettes, taken over all…
The degree/diameter problem for mixed graphs asks for the largest possible order of a mixed graph with given diameter and degree parameters. Similarly the \emph{degree/geodecity} problem concerns the smallest order of a $k$-geodetic mixed…
A \emph{Stick graph} is an intersection graph of axis-aligned segments such that the left end-points of the horizontal segments and the bottom end-points of the vertical segments lie on a `ground line,' a line with slope $-1$. It is an open…
An oriented k-uniform hypergraph (a family of ordered k-sets) has the ordering property (or Property O) if for every linear order of the vertex set, there is some edge oriented consistently with the linear order. We find bounds on the…
A linear coloring of a graph is a proper coloring of the vertices of the graph so that each pair of color classes induce a union of disjoint paths. In this paper, we prove that for every connected graph with maximum degree at most three and…
An ordered hypergraph is a hypergraph $H$ with a specified linear ordering of the vertices, and the appearance of an ordered hypergraph $G$ in $H$ must respect the specified order on $V(G)$. In on-line Ramsey theory, Builder iteratively…