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View article: Solution to some conjectures on mobile position problems
Solution to some conjectures on mobile position problems Open
The general position problem for graphs asks for the largest number of vertices in a subset $S \subseteq V(G)$ of a graph $G$ such that for any $u,v \in S$ and any shortest $u,v$-path $P$ we have $S \cap V(P) = \{ u,v\} $, whereas the mutu…
View article: The General Position Problem: A Survey
The General Position Problem: A Survey Open
Inspired by a chessboard puzzle of Dudeney, the general position problem in graph theory asks for a largest set $S$ of vertices in a graph such that no three elements of $S$ lie on a common shortest path. The number of vertices in such a l…
View article: On bipartite (1,1,k)-mixed graphs
On bipartite (1,1,k)-mixed graphs Open
Mixed graphs can be seen as digraphs that have both arcs and edges (or digons, that is, two opposite arcs). In this paper, we consider the case where such graphs are bipartite and in which the undirected and directed degrees are one. The b…
View article: Monophonic position sets of Cartesian and lexicographic products of graphs
Monophonic position sets of Cartesian and lexicographic products of graphs Open
The general position problem in graph theory asks for the number of vertices in a largest set $S$ of vertices of a graph $G$ such that no shortest path of $G$ contains more than two vertices of $S$. The analogous monophonic position proble…
View article: Colouring a graph with position sets
Colouring a graph with position sets Open
In this paper we consider a colouring version of the general position problem. The \emph{$\gp $-chromatic number} is the smallest number of colours needed to colour the vertices of the graph such that each colour class has the no-three-in-…
View article: Lower General Position in Cartesian Products
Lower General Position in Cartesian Products Open
A subset $S$ of vertices of a graph $G$ is in \emph{general position} if no shortest path in $G$ contains three vertices of $S$. The \emph{general position problem} consists of finding the number of vertices in a largest general position s…
View article: On the general position number of Mycielskian graphs
On the general position number of Mycielskian graphs Open
The general position problem for graphs was inspired by the no-three-in-line problem from discrete geometry. A set S of vertices of a graph G is a general position set if no shortest path in G contains three or more vertices of S. The gene…
View article: On bipartite $(1,1,k)$-mixed graphs
On bipartite $(1,1,k)$-mixed graphs Open
Mixed graphs can be seen as digraphs with arcs and edges (or digons, that is, two opposite arcs). In this paper, we consider the case where such graphs are bipartite and in which the undirected and directed degrees are one. The best graphs…
View article: Lower general position sets in graphs
Lower general position sets in graphs Open
A subset S of vertices of a graph G is a general position set if no shortest path in G contains three or more vertices of S. In this paper, we generalise a problem of M. Gardner to graph theory by introducing the lower general position num…
View article: The structure of digraphs with excess one
The structure of digraphs with excess one Open
A digraph is ‐ geodetic if for any (not necessarily distinct) vertices there is at most one directed walk from to with length not exceeding . The order of a ‐geodetic digraph with minimum out‐degree is bounded below by the directed Moore b…
View article: General position polynomials
General position polynomials Open
A subset of vertices of a graph $G$ is a general position set if no triple of vertices from the set lie on a common shortest path in $G$. In this paper we introduce the general position polynomial as $\sum_{i \geq 0} a_i x^i$, where $a_i$ …
View article: Builder-Blocker General Position Games
Builder-Blocker General Position Games Open
This paper considers a game version of the general position problem in which a general position set is built through adversarial play. Two players in a graph, Builder and Blocker, take it in turns to add a vertex to a set, such that the ve…
View article: On some extremal position problems for graphs
On some extremal position problems for graphs Open
The general position number of a graph G is the size of the largest set of vertices S such that no geodesic of G contains more than two elements of S. The monophonic position number of a graph is defined similarly, but with `induced path' …
View article: Lower General Position Sets in Graphs
Lower General Position Sets in Graphs Open
A subset $S$ of vertices of a graph $G$ is a \emph{general position set} if no shortest path in $G$ contains three or more vertices of $S$. In this paper, we generalise a problem of M. Gardner to graph theory by introducing the \emph{lower…
View article: On large regular (1,1,k)-mixed graphs
On large regular (1,1,k)-mixed graphs Open
An $(r,z,k)$-mixed graph $G$ has every vertex with undirected degree $r$, directed in- and out-degree $z$, and diameter $k$. In this paper, we study the case $r=z=1$, proposing some new constructions of $(1,1,k)$-mixed graphs with a large …
View article: Mutually avoiding Eulerian circuits
Mutually avoiding Eulerian circuits Open
Two Eulerian circuits, both starting and ending at the same vertex, are avoiding if at every other point of the circuits they are at least distance 2 apart. An Eulerian graph which admits two such avoiding circuits starting from any vertex…
View article: Small Graphs and Hypergraphs of Given Degree and Girth
Small Graphs and Hypergraphs of Given Degree and Girth Open
The search for the smallest possible $d$-regular graph of girth $g$ has a long history, and is usually known as the cage problem. This problem has a natural extension to hypergraphs, where we may ask for the smallest number of vertices in …
View article: On the vertex position number of graphs
On the vertex position number of graphs Open
In this paper we generalise the notion of visibility from a point in an integer lattice to the setting of graph theory.For a vertex x of a graph G, we say that a set S ⊆ V (G) is an x-position set if for any y ∈ S the shortest 2 Thankachy,…
View article: On monophonic position sets in graphs
On monophonic position sets in graphs Open
The general position problem in graph theory asks for the largest set S of vertices of a graph G such that no shortest path of G contains more than two vertices of S. In this paper we consider a variant of the general position problem call…
View article: The maximum Wiener index of a uniform hypergraph
The maximum Wiener index of a uniform hypergraph Open
The Wiener index of a (hyper)graph is calculated by summing up the distances between all pairs of vertices. We determine the maximum possible Wiener index of a connected $n$-vertex $k$-uniform hypergraph and characterize for every~$n$ all …
View article: TRAVERSING A GRAPH IN GENERAL POSITION
TRAVERSING A GRAPH IN GENERAL POSITION Open
Let G be a graph. Assume that to each vertex of a set of vertices $S\subseteq V(G)$ a robot is assigned. At each stage one robot can move to a neighbouring vertex. Then S is a mobile general position set of G if there exists a sequence of …
View article: Traversing a graph in general position
Traversing a graph in general position Open
Let $G$ be a graph. Assume that to each vertex of a set of vertices $S\subseteq V(G)$ a robot is assigned. At each stage one robot can move to a neighbouring vertex. Then $S$ is a mobile general position set of $G$ if there exists a sequen…
View article: On the Vertex Position Number of Graphs
On the Vertex Position Number of Graphs Open
In this paper we generalise the notion of visibility from a point in an integer lattice to the setting of graph theory. For a vertex $x$ of a connected graph $G$, we say that a set $S \subseteq V(G)$ is an \emph{$x$-position set} if for an…
View article: On Networks with Order Close to the Moore Bound
On Networks with Order Close to the Moore Bound Open
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 degree/geodecity problem concerns the smallest order of a k -geodetic mixed graph wi…
View article: On the General Position Number of Mycielskian Graphs
On the General Position Number of Mycielskian Graphs Open
The general position problem for graphs was inspired by the no-three-in-line problem from discrete geometry. A set $S$ of vertices of a graph $G$ is a \emph{general position set} if no shortest path in $G$ contains three or more vertices o…
View article: Small graphs and hypergraphs of given degree and girth
Small graphs and hypergraphs of given degree and girth Open
The search for the smallest possible $d$-regular graph of girth $g$ has a long history, and is usually known as the cage problem. This problem has a natural extension to hypergraphs, where we may ask for the smallest number of vertices in …
View article: The structure of digraphs with excess one
The structure of digraphs with excess one Open
A digraph $G$ is \emph{$k$-geodetic} if for any (not necessarily distinct) vertices $u,v$ there is at most one directed walk from $u$ to $v$ with length not exceeding $k$. The order of a $k$-geodetic digraph with minimum out-degree $d$ is …