一个\textit{diameter graph in $\mathbb R^d$}是一个图，其顶点集是$\mathbb R^d$的有限子集，其边集是由在直径上分开的顶点对形成的。本文研究了直径图在$\mathbb R^4$中和三维球面上的不同极值性质。对于半径大于$1/\sqrt 2$的三维球面上的直径图，我们证明了Vázsonyi和Borsuk猜想的一个类似结果.证明了$\mathbb R^4.$中直径图的Schur猜想。我们还建立了$\mathbb R^4$中直径图所能具有的最大三角形数，证明了极值只在特定的Lenz构型上才能达到。
英文摘要：
A \textit{diameter graph in $\mathbb R^d$} is a graph, whose set of vertices is a finite subset of $\mathbb R^d$ and whose set of edges is formed by pairs of vertices that are at diameter apart. This paper is devoted to the study of different extremal properties of diameter graphs in $\mathbb R^4$ and on a three-dimensional sphere. We prove an analogue of Vázsonyi's and Borsuk's conjecture for diameter graphs on a three-dimensional sphere with radius greater than $1/\sqrt 2$. We prove Schur's conjecture for diameter graphs in $\mathbb R^4.$ We also establish the maximum number of triangles a diameter graph in $\mathbb R^4$ can have, showing that the extremum is attained only on specific Lenz configurations.
In 1975, Plesník characterized all triangle-free planar graphs as having a diameter $2$. We characterize all triangle-free projective-planar graphs having a diameter $2$ and discuss some applications. In particular, the main result is applied to calculate the analogue of clique numbers for graphs, namely, colored mixed graphs, having different types of arcs and edges.
The game of Cops and Robbers is a well known game played on graphs. In this paper we consider the class of graphs of bounded diameter. We improve the strategy of cops and previously used probabilistic method which results in an improved upper bound for the cop number of graphs of bounded diameter. In particular, for graphs of diameter four, we improve the upper bound from $n^{\frac{2}{3}+o(1)}$ to $n^{\frac{3}{5}+o(1)}$ and for diameter three from $n^{\frac{2}{3}+o(1)}$ to $n^{\frac{4}{7}+o(1)}$.
A finite set $A \subset \mathbb{R}^d$ is called $\textit{diameter-Ramsey}$ if for every $r \in \mathbb N$, there exists some $n \in \mathbb N$ and a finite set $B \subset \mathbb{R}^n$ with $\mathrm{diam}(A)=\mathrm{diam}(B)$ such that whenever $B$ is coloured with $r$ colours, there is a monochromatic set $A' \subset B$ which is congruent to $A$. We prove that sets of diameter $1$ with circumradius larger than $1/\sqrt{2}$ are not diameter-Ramsey. In particular, we obtain that triangles with an angle larger than $135^\circ$ are not diameter-Ramsey, improving a result of Frankl, Pach, Reiher a…
Graph associahedra are generalized permutohedra arising as special cases of nestohedra and hypergraphic polytopes. The graph associahedron of a graph $G$ encodes the combinatorics of search trees on $G$, defined recursively by a root $r$ together with search trees on each of the connected components of $G-r$. In particular, the skeleton of the graph associahedron is the rotation graph of those search trees. We investigate the diameter of graph associahedra as a function of some graph parameters. We give a tight bound of $Θ(m)$ on the diameter of trivially perfect graph associahedra on $m$ edge…
An orientation of an undirected graph $G$ is an assignment of exactly one direction to each edge of $G$. The oriented diameter of a graph $G$ is the smallest diameter among all the orientations of $G$. The maximum oriented diameter of a family of graphs $\mathscr{F}$ is the maximum oriented diameter among all the graphs in $\mathscr{F}$. Chvátal and Thomassen [JCTB, 1978] gave a lower bound of $\frac{1}{2}d^2+d$ and an upper bound of $2d^2+2d$ for the maximum oriented diameter of the family of $2$-edge connected graphs of diameter $d$. We improve this upper bound to $ 1.373 d^2 + 6.971d-1 $, w…
Using the domination parameters of Cayley graphs constructed out of $\mathbb{Z}_{p}\times \mathbb{Z}_{m}$, where $m\in\{p^α, p^αq^β, p^αq^βr^γ\},$ in this paper we are discussing about the total and connected domination number and diameter of these Cayley graphs.
In this paper we introduce Yoke graphs, a family of flip graphs that generalizes several previously studied families of graphs: colored triangle free triangulations, arc permutations and caterpillars. Our main result is the computation of the diameter of an arbitrary Yoke graph.
Almost Moore mixed graphs\/} appear in the context of the degree/diameter problem as a class of extremal mixed graphs, in the sense that their order is one unit less than the Moore bound for such graphs. The problem of their existence has been considered just for diameter $2$. In this paper, we give a complete characterization of these extremal mixed graphs for diameters 2 and 3. We also derive some optimal constructions for other diameters.
The degree-diameter problem seeks to find the largest possible number of vertices in a graph having given diameter and given maximum degree. Very often the problem is studied for restricted families of graph such as vertex-transitive or Cayley graphs, with the goal being to find a family of graphs with good asymptotic properties. In this paper we restrict attention to Cayley graphs, and study the asymptotics by fixing a small diameter and constructing families of graphs of large order for all values of the maximum degree. Much of the literature in this direction is focused on the diameter two …
We obtain a new bound connecting the first non--trivial eigenvalue of the Laplace operator of a graph and the diameter of the graph, which is effective for graphs with small diameter or for graphs, having the number of maximal paths comparable to the expectation.
We consider the localization number and metric dimension of certain graphs of diameter $2$, focusing on families of Kneser graphs and graphs without 4-cycles. For the Kneser graphs with diameter $2$, we find upper and lower bounds for the localization number and metric dimension, and in many cases these parameters differ only by an additive constant. Our results on the metric dimension of Kneser graphs improve on earlier ones, yielding exact values in infinitely many cases. We determine bounds on the localization number and metric dimension of Moore graphs of diameter $2$ and polarity graphs.
This paper considers the degree-diameter problem for undirected circulant graphs. For degrees 10 and 11 newly discovered families of circulant graphs of arbitrary diameter are presented which are largest known and are conjectured to be extremal. They are also the largest-known Abelian Cayley graphs of these degrees. For each such family the order of every graph in the family is defined by a quintic polynomial function of the diameter which is specific to the family. The elements of the generating set for each graph are similarly defined by a set of polynomials in the diameter. The existence of…
Cubic planar $n$-vertex graphs with faces of length at most $6$, e.g., fullerene graphs, have diameter in $Ω(\sqrt{n})$. It has been suspected, that a similar result can be shown for cubic planar graphs with faces of bounded length. This note provides a family of cubic planar $n$-vertex graphs with faces of length at most $7$ and diameter in ${O}(\log n)$, thus refuting the above suspicion.
A consequence of Ore's classic theorem characterizing the maximal graphs with given order and diameter is a determination of the largest such graphs. We give a very short and simple proof of this smaller result, based on a well-known elementary observation.
The $α$-spectral radius of a connected graph $G$ is the spectral radius of $A_α$-matrix of $G$. In this paper, we discuss the methods for comparing $α$-spectral radius of graphs. As applications, we characterize the graphs with the maximal $α$-spectral radius among all unicyclic and bicyclic graphs of order $n$ with diameter $d$, respectively. Finally, we determine the unique graph with maximal signless Laplacian spectral radius among bicyclic graphs of order $n$ with diameter $d$.
The Steiner $k$-diameter $sdiam_k(G)$ of a graph $G$, introduced by Chartrand, Oellermann, Tian and Zou in 1989, is a natural generalization of the concept of classical diameter. When $k=2$, $sdiam_2(G)=diam(G)$ is the classical diameter. The problem of determining the minimum size of a graph of order $n$ whose diameter is at most $d$ and whose maximum is $\ell$ was first introduced by Erdös and Rényi. In this paper, we generalize the above problem for Steiner $k$-diameter, and study the problem of determining the minimum size of a graph of order $n$ whose Steiner $3$-diameter is at most $d$ a…
It was conjectured by Koh and Tay [Graphs Combin. 18(4) (2002), 745--756] that for $n\geq 5$ every simple graph of order $n$ and size at least $\binom{n}{2}-n+5$ has an orientation of diameter two. We prove this conjecture and hence determine for every $n\geq 5$ the minimum value of $m$ such that every graph of order $n$ and size $m$ has an orientation of diameter two.
We prove that a graph has an r-bounded subdivision of a wheel if and only if it does not have a graph-decomposition of locality r and width at most two.
Let $G$ be a graph of radius $r$ and diameter $d$ with $d\leq 2r-2$. We give a new proof that $G$ contains a cycle of length at least $4r-2d$, i.e. for its circumference it holds $c(G)\geq 4r-2d$.
For $n \in \mathbb{N}$ let $δ_n$ be the smallest value such that every graph of order $n$ and minimum degree at least $δ_n$ admits an orientation of diameter two. We show that $δ_n=\frac{n}{2} + Θ(\ln n)$.
We study bipartite distance-regular Cayley graphs with diameter three or four. We give sufficient conditions under which a bipartite Cayley graph can be constructed on the semidirect product of a group -- the part of this bipartite Cayley graph which contains the identity element -- and $\mathbb{Z}_{2}$. We apply this to the case of bipartite distance-regular Cayley graphs with diameter three, and consider cases where the sufficient conditions are not satisfied for some specific groups such as the dihedral group. We also extend a result by Miklavič and Potočnik that relates difference sets to …
DP-coloring (also known as correspondence coloring) is a generalization of list coloring introduced by Dvourák and Postle (2017). Recently, Huang et al. [https://doi.org/10.1016/j.amc.2019.124562] showed that planar graphs with diameter at most two are $4$-choosable. In this paper, we will prove that planar graphs with diameter at most two are DP-$4$-colorable, which is an extension of the above result.
Regular graphs which are Bonnet-Myers sharp (in the sense of Ollivier Ricci curvature) and self-centered have been completely classified, and it is a natural question whether the condition of self-centeredness can be removed in the classification. We prove that this condition is indeed not necessary in the special case of Bonnet-Myers sharp graphs of diameter 3.
