In this paper, we study the oriented diameter of power graphs of groups. We show that a $2$-edge connected power graph of a finite group has oriented diameter at most $4$. We prove that the power graph of the cyclic group of order $n$ has oriented diameter $2$ for all $n\neq 1,2,4,6$. For non-cyclic finite nilpotent groups, we show that the oriented diameter of corresponding power graphs is at least $3$. Moreover, we provide necessary and sufficient conditions for the oriented diameter of $2$-edge connected power graphs of finite non-cyclic nilpotent groups to be either $3$ or $4$. This, in turn, gives an algorithm for computing the oriented diameter of the power graph of a given nilpotent group that runs in time polynomial in the size of the group.
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…
We consider the problem of finding an orientation with minimum diameter of a connected bridgeless graph. Fomin et. al. discovered a relation between the minimum oriented diameter an the size of a minimal dominating set. We improve their upper bound.
In 1978, Chvátal and Thomassen showed that every bridgeless graph with diameter 2 has an orientation with diameter at most 6. They also gave general bounds on the smallest value $f(d)$ such that every bridgeless graph $G$ with diameter $d$ has an orientation with diameter at most $f(d)$. For $d=3$, they proved that $8\leq f(d)\leq 24$. Until recently, Kwok, Liu and West improved the above bounds by proving $9\leq f(3)\leq 11$ in [P.K. Kwok, Q. Liu and D.B. West, Oriented diameter of graphs with diameter 3, J. Combin. Theory Ser.B 100(2010), 265-274]. In this paper, we determine the oriented di…
Oriented matroids (often called order types) are combinatorial structures that generalize point configurations, vector configurations, hyperplane arrangements, polyhedra, linear programs, and directed graphs. Oriented matroids have played a key role in combinatorics, computational geometry, and optimization. This paper surveys prior work and presents an update on the search for bounds on the diameter of the cocircuit graph of an oriented matroid. We review the diameter problem and show the diameter bounds of general oriented matroids reduce to those of uniform oriented matroids. We give the la…
Embracing exchange sequences and oriented matroid polyhedron diameter
MTA-ELTE Matroid Optimization Research Group and HUN-REN-ELTE Egerváry Research Group, Department of Operations Research, ELTE Eötvös Loránd University（匈牙利科学院-罗兰大学拟阵优化研究组和匈牙利研究与创新网络-罗兰大学埃格瓦里研究组，运筹学系，罗兰大学）；HUN-REN Alfréd Rényi Institute of Mathematics（匈牙利研究与创新网络阿尔弗雷德·雷尼数学研究所）；Departament de Matemàtiques i Informàtica, Universitat de Barcelona, Centre de Recerca Matemàtica（巴塞罗那大学数学与计算机科学系，数学研究中心）；Departament d’Enginyeria Informàtica i Matemàtiques, Universitat Rovira i Virgili（加泰罗尼亚理工大学生物信息学与数学工程系）
Let $S(G^σ)$ be the skew-adjacency matrix of the oriented graph $G^σ$, which is obtained from a simple undirected graph $G$ by assigning an orientation $σ$ to each of its edges. The skew energy of an oriented graph $G^σ$ is defined as the sum of absolute values of all eigenvalues of $S(G^σ)$. For any positive integer $d$ with $3\leq d\leq n-3$, we determine the graph with minimal skew energy among all oriented bicyclic graphs that contain no vertex disjoint odd cycle of lengths $s$ and $l$ with $s+l\equiv 2(mod 4)$ on $n$ vertices with a given diameter $d$.
The oriented diameter of a bridgeless graph $G$ is $\min\{diam(H)\ | H\ is\ an orientation\ of\ G\}$. A path in an edge-colored graph $G$, where adjacent edges may have the same color, is called rainbow if no two edges of the path are colored the same. The rainbow connection number $rc(G)$ of $G$ is the smallest integer $k$ for which there exists a $k$-edge-coloring of $G$ such that every two distinct vertices of $G$ are connected by a rainbow path. In this paper, we obtain upper bounds for the oriented diameter and the rainbow connection number of a graph in terms of $rad(G)$ and $η(G)$, wher…
In 2015, Dankelmann and Bau proved that for every bridgeless graph $G$ of order $n$ and minimum degree $δ$ there is an orientation of diameter at most $11\frac{n}{δ+1}+9$. In 2016, Surmacs reduced this bound to $7\frac{n}{δ+1}.$ In this paper, we consider the girth of a graph $g$ and show that for any $\varepsilon>0$ there is a bound of the form $(2g+\varepsilon)\frac{n}{h(δ,g)}+O(1)$, where $h(δ,g)$ is a polynomial. Letting $g=3$ and $\varepsilon<1$ gives an inprovement on the result by Surmacs.
{\em Arrowhead} and {\em diamond} are two hierarchical Cayley graphs defined on the triangular grid. In their undirected version, they are isomorphic and merely define two distinct representations of the same graph. This paper gives the expression of their diameter, in the oriented and non--oriented case. It also displays the full distribution of antipodals.
Assignment of one of the two possible directions to every edge of an undirected graph $G=(V,E)$ is called an orientation of $G$. The resulting directed graph is denoted by $\overrightarrow{G}$. A strong orientation is one in which every vertex is reachable from every other vertex via a directed path. The diameter of $\overrightarrow{G}$, i.e., the maximum distance from one vertex to another, depends on the particular orientation. The minimum diameter among all possible orientations is called the oriented diameter $\overrightarrow{\text{diam}}(G)$ of $G$. Let $n,k$ be two integers with $1 \leq …
An {\em orientation} of an undirected graph $G$ is an assignment of exactly one direction to each edge of $G$. Converting two-way traffic networks to one-way traffic networks and bidirectional communication networks to unidirectional communication networks are practical instances of graph orientations. In these contexts minimising the diameter of the resulting oriented graph is of prime interest. The $n$-star network topology was proposed as an alternative to the hypercube network topology for multiprocessor systems by Akers and Krishnamurthy [IEEE Trans. on Computers (1989)]. The $n$-star gra…
An asymptotically tight bound on oriented diameter
Nankai University（南开大学）；Nanjing University（南京大学）；The Hong Kong Polytechnic University（香港理工大学）；East China Normal University（华东师范大学）；Fuzhou University（福州大学）
Inversion diameter and 2-edge-colored homomorphisms
Université Paris Cité（巴黎西岱大学）；CNRS（法国国家科学研究中心）；IRIF（IRIF实验室）；Sorbonne Université（索邦大学）；LIP6（LIP6实验室）；Centre d’Analyse et de Mathématique Sociales（社会分析与数学中心）；EHESS（高等社会科学研究院）；LISN（LISN实验室）；Université Paris-Saclay（巴黎萨克雷大学）；Univ Lyon（里昂大学）；UCBL（里昂第一大学）；INSA Lyon（里昂国立应用科学学院）；LIRIS（LIRIS实验室）；University of Warsaw（华沙大学）
By using a $q$-analogue of the "magic" matrix introduced by H.Huang in his elegant solution of the sensitivity conjecture, we give a direct generalization of his result, replacing a hypercube graph by a Cartesian power of a directed $l$-cycle.
A computer search through the oriented matroid programs with dimension 5 and 10 facets shows that the maximum strictly monotone diameter is 5. Thus $Δ_{sm}(5,10)=5$. This enumeration is analogous to that of Bremner and Schewe for the non-monotone diameter of 6-polytopes with 12 facets. Similar enumerations show that $Δ_{sm}(4,9)=5$ and $Δ_m(4,9)=Δ_m(5,10)=6.$ We shorten the known non-computer proof of the strict monotone 4-step conjecture.
Let $\mathcal{D}$ be a weighted oriented graph and $I(\mathcal{D})$ be its edge ideal. In this paper, we show that all the symbolic and ordinary powers of $I(\mathcal{D})$ coincide when $\mathcal{D}$ is a weighted oriented certain class of tree. Finally, we give necessary and sufficient conditions for the equality of ordinary and symbolic powers of naturally oriented lines.
We consider two orientation problems in a graph, namely the minimization of the sum of all the shortest path lengths and the minimization of the diameter. We show that it is NP-complete to decide whether a graph has an orientation such that the sum of all the shortest paths lengths is at most an integer specified in the input. The proof is a short reduction from a result of Chvátal and Thomassen showing that it is NP-complete to decide whether a graph can be oriented so that its diameter is at most 2. In contrast, for each positive integer k, we describe a linear-time algorithm that decides fo…
Pebbling is a game played on a graph. The single player is given a graph and a configuration of pebbles and may make pebbling moves by removing 2 pebbles from one vertex and placing one at an adjacent vertex to eventually have one pebble reach a predetermined vertex. The pebbling number, $π(G)$, is the minimum number of pebbles such that regardless of their exact configuration, the player can use pebbling moves to have a pebble reach any predetermined vertex. Previous work has related $π(G)$ to the diameter of $G$. Clarke, Hochberg, and Hurlbert demonstrated that every connected undirected gra…
The diameter and dominating sets of the difference graph of a nilpotent group
School of Science, Xi’an Shiyou University（西安石油大学理学院）；University of Novi Sad, Faculty of Sciences, Department of Mathematics and Informatics（诺维萨德大学理学院）；Mathematical Institute of the Serbian Academy of Sciences and Arts（塞尔维亚科学与艺术学院数学研究所）
In this paper, we investigate the diameter in preferential attachment (PA-) models, thus quantifying the statement that these models are small worlds. The models studied here are such that edges are attached to older vertices proportional to the degree plus a constant, i.e., we consider affine PA-models. There is a substantial amount of literature proving that, quite generally, PA-graphs possess power-law degree sequences with a power-law exponent τ>2. We prove that the diameter of the PA-model is bounded above by a constant times \log{t}, where t is the size of the graph. When the power-law e…
Improving a recent result of Fundikwa, Mazorodze, and Mukwembi, we show that $d \leq (2n-3)/5$ for every connected $C_4$-free graph of order $n$, diameter $d$, and edge-connectivity at least $3$, which is best possible up to a small additive constant. For edge-connectivity at least $4$, we improve this to $d \leq (n-3)/3$. Furthermore, adapting a construction due to Erdős, Pach, Pollack, and Tuza, for an odd prime power $q$ at least $7$, and every positive integer $k$, we show the existence of a connected $C_4$-free graph of order $n=(q^2+q-1)k+1$, diameter $d=4k$, and edge-connectivity $λ$ at…
It is known that a distance-regular graph with valency $k$ at least three admits at most two Q-polynomial structures. % In this note we show that all distance-regular graphs with diameter four and valency at least three admitting two $Q$-polynomial structures are either dual bipartite or almost dual imprimitive. By the work of Dickie \cite{Dickie} this implies that any distance-regular graph with diameter $d$ at least four and valency at least three admitting two $Q$-polynomial structures is, provided it is not a Hadamard graph, either the cube $H(d,2)$ with $d$ even, the half cube ${1}/{2} H(…
We consider bicolored maps, i.e. graphs which are drawn on surfaces, and construct a bijection between (i) oriented maps with arbitary face structure, and (ii) (weighted) non-oriented maps with exactly one face. Above, each non-oriented map is counted with a multiplicity which is based on the concept of the orientability generating series and the measure of orientability of a map. This bijection has the remarkable property of preserving the underlying bicolored graph. Our bijection shows equivalence between two explicit formulas for the top-degree of Jack characters, i.e. (suitably normalized)…
For every $n\in\mathbb N$ we construct a finite graph $G$ such that every orientation $\vec G$ of $G$ contains an isometric copy of any oriented tree on $n$ vertices, and evaluate the smallest possible cardinality of $G$. On the other hand, we prove that every graph $G$ admits an orientation containing no directed $ω$-paths of infinite diameter.
A particular case of Caccetta-Häggkvist conjecture, says that a digraph of order $n$ with minimum out-degree at least $1/3n$ contains a directed cycle of length at most 3. Recently, Kral, Hladky and Norine proved that a digraph of order $n$ with minimum out-degree at least $0.3465n$ contains a directed cycle of length at most 3 (which currently is the best result). A weaker particular case says that a digraph of order $n$ with minimum semi-degree at least $1/3n$ contains a directed triangle. In a recent paper, by using the result of Kral et al, the author proved that for $β\geq 0.343545$, any …
Let $Γ$ be a group and $(Γ_n)_{n=1} ^{\infty}$ be a descending sequence of finite-index normal subgroups. We establish explicit upper bounds on the diameters of the directed Cayley graphs of the $Γ/Γ_n$ , under some natural hypotheses on the behaviour of power and commutator words in $Γ$. The bounds we obtain do not depend on a choice of generating set. Moreover under reasonable conditions our method provides a fast algorithm for constructing directed Cayley graphs of diameter satisfying our bounds. The proof is closely analogous to the the Solovay-Kitaev procedure, which only uses commutator …
Bernoulli Institute, Groningen University（格罗宁根大学伯努利研究所）；Department of Ecology and Evolutionary Biology, Princeton University（普林斯顿大学生态与进化生物学系）；Utrecht Uniersity（乌得勒支大学）
The zero forcing number is the minimum number of black vertices that can turn a white graph black following a single neighbour colour forcing rule. The zero forcing number provides topological information about linear algebra on graphs, with applications to the controllability of linear dynamical systems and quantum walks on graphs among other problems. Here, I investigate the zero forcing number of undirected simple graphs with a power law degree distribution $p_k\sim k^{-γ}$. For graphs generated by the preferential attachment model, with a diameter scaling logarithmically with the graph siz…
In this paper we study irreducible representations and symbolic Rees algebras of monomial ideals. Then we examine edge ideals associated to vertex-weighted oriented graphs. These are digraphs having no oriented cycles of length two with weights on the vertices. For a monomial ideal with no embedded primes we classify the normality of its symbolic Rees algebra in terms of its primary components. If the primary components of a monomial ideal are normal, we present a simple procedure to compute its symbolic Rees algebra using Hilbert bases, and give necessary and sufficient conditions for the equ…
