Marcel Koloschin, Max Pitz
Building on recent work by Thomassen, we show that Nash-Williams' orientation theorem, that every finite $2k$-edge-connected multigraph has a $k$-arc-connected orientation, also holds for all infinite multigraphs.
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Towards Nash-Williams Orientation Conjecture for Infinite Graphs
本文研究了无限图的Nash-Williams定向猜想，改进了Thomassen 2016年的结果，证明对于单端局部有限图，4k边连通性足以保证k-弧连通定向，关键依赖于提升图概念的新结果。
The strong Nash-Williams orientation theorem for rayless graphs
针对Nash-Williams提出的无限图也存在满足强定向定理的定向方式的猜想，本研究证明所有无射线图都符合该定向要求，为该猜想提供了部分解答。
We study graphs which admit an acyclic orientation that contains an out-branching and in-branching which are arc-disjoint (such an orientation is called {\bf good}). A {\bf 2T-graph} is a graph whose edge set can be decomposed into two edge-disjoint spanning trees. Clearly a graph has a good orientation if and only if it contains a spanning 2T-graph with a good orientation, implying that 2T-graphs play a central role. Vertex-minimal 2T-graphs with at least two vertices, also known as {\bf generic circuits}, play an important role in rigidity theory for graphs. It was shown in \cite{bangGOpaper…
Every connected graph admits a local antimagic orientation and almost every graph admits an antimagic orientation
本文证明了每个连通图都存在局部反魔方向，并证明几乎所有图都存在反魔方向，从而肯定回答了Chang等人和Hefetz等人的相关猜想。
In this short notice we give a universal definition of $\Z_2$-module $Or(Γ)$ of orientations of a graph $Γ$ and construct a method, by means of which one can easily verify whenever two such special definitions coincide.
Proper 3-orientations of bipartite planar graphs with minimum degree at least 3
研究证明，最小度数至少为3的双部平面图的恰当定向数不超过3。
In this short note we determine the maximum number, over all $n$-vertex graphs $G$, of orientations of $G$ containing no strongly connected cycle $C_{2k+1}$. This answers a part of a recent question of Araujo, Botler and Mota.
A simple graph $G$ is said to admit an antimagic orientation if there exist an orientation on the edges of $G$ and a bijection from $E(G)$ to $\{1,2,\ldots,|E(G)|\}$ such that the vertex sums of vertices are pairwise distinct, where the vertex sum of a vertex is defined to be the sum of the labels of the in-edges minus that of the out-edges incident to the vertex. It was conjectured by Hefetz, Mütze, and Schwartz~\cite{HMS10} in 2010 that every connected simple graph admits an antimagic orientation. In this paper, we prove that the Mycielski construction and the corona product for graphs with …
An antimagic labeling of a directed graph $D$ with $n$ vertices and $m$ arcs is a bijection from the set of arcs of $D$ to the integers $\{1, \cdots, m\}$ such that all $n$ oriented vertex sums are pairwise distinct, where an oriented vertex sum is the sum of labels of all arcs entering that vertex minus the sum of labels of all arcs leaving it. A graph $G$ has an antimagic orientation if it has an orientation which admits an antimagic labeling. Hefetz, M{ü}tze, and Schwartz conjectured that every connected graph admits an antimagic orientation. In this paper, we show that every bipartite grap…
Motivated by the conjecture of Hartsfield and Ringel on antimagic labelings of undirected graphs, Hefetz, Mütze, and Schwartz initiated the study of antimagic labelings of digraphs in 2010. Very recently, it has been conjectured in [Antimagic orientation of even regular graphs, J. Graph Theory, 90 (2019), 46-53.] that every graph admits an antimagtic orientation, which is a strengthening of an earlier conjecture of Hefetz, Mütze and Schwartz. In this paper, we prove that every $2d$-regular graph (not necessarily connected) admits an antimagic orientation, where $d\ge2$. Together with known res…
An antimagic labeling of a directed graph $D$ with $n$ vertices and $m$ arcs is a bijection from the set of arcs of $D$ to the integers $\{1, \cdots, m\}$ such that all $n$ oriented vertex sums are pairwise distinct, where an oriented vertex sum is the sum of labels of all arcs entering that vertex minus the sum of labels of all arcs leaving it. An undirected graph $G$ is said to have an antimagic orientation if $G$ has an orientation which admits an antimagic labeling. Hefetz, M{ü}tze, and Schwartz conjectured that every connected undirected graph admits an antimagic orientation. In this pape…
We give a characterization of Pfaffian graphs in terms of even orientations, extending the characterization of near bipartite non--pfaffian graphs by Fischer and Little \cite{FL}. Our graph theoretical characterization is equivalent to the one proved by Little in \cite{L73} (cf. \cite{LR}) using linear algebra arguments.
An orientation of a graph $G$ is {\it in-out-proper} if any two adjacent vertices have different in-out-degrees, where the in-out-degree of each vertex is equal to the in-degree minus the out-degree of that vertex. The {\it in-out-proper orientation number} of a graph $G$, denoted by $\overleftrightarrowχ(G)$, is $ \min_{D\in Γ}\max_{v\in V(G)} |d_D^{\pm}(v)|$, where $Γ$ is the set of in-out-proper orientations of $G$ and $d_D^{\pm}(v)$ is the in-out-degree of the vertex $v$ in the orientation $D$.
We use a randomised embedding method to prove that for all α>0 any sufficiently large oriented graph G with minimum in-degree and out-degree δ^+(G),δ^-(G)\geq (3/8+α)|G| contains every possible orientation of a Hamilton cycle. This confirms a conjecture of Häggkvist and Thomason.
We extend the duality between acyclic orientations and totally cyclic orientations on planar graphs to dualities on graphs on orientable surfaces by introducing boundary acyclic orientations and totally bi-walkable orientations. In addition, we provide a reciprocity theorem connecting local tensions and boundary acyclic orientations. Furthermore, we define the balanced flow polynomial which is connected with tension polynomial by duality and with totally bi-walkable orientations by reciprocity.
The notion of a Galvin orientation of a line graph is introduced, generalizing the idea used by Galvin in his landmark proof of the list-edge-colouring conjecture for bipartite graphs. If L(G) has a proper Galvin orientation with respect to k, then it immediately implies that G is k-list-edge-colourable, but the converse is not true. The stronger property is studied in graphs of the form `bipartite plus an edge', the Petersen graph, cliques, and simple graphs without odd cycles of length 5 or longer.
Felsner introduced a cycle reversal, namely the `flip' reversal, for α-orientations (i.e., each vertex admits a prescribed out-degree) of a graph G embedded on the plane and further proved that the set of all the α-orientations of G carries a distributive lattice with respect to the flip reversals. In this paper, we give an explicit formula for the minimum number of flips needed to transform one α-orientation into another for graphs embedded on the plane or sphere, respectively.
An orientation of a graph $G$ is proper if any two adjacent vertices have different indegrees. The proper orientation number $\overrightarrowχ(G)$ of a graph $G$ is the minimum of the maximum indegree, taken over all proper orientations of $G$. In this paper, we show that a connected bipartite graph may be properly oriented even if we are only allowed to control the orientation of a specific set of edges, namely, the edges of a spanning tree and all the edges incident to one of its leaves. As a consequence of this result, we prove that 3-connected planar bipartite graphs have proper orientatio…
Antimagic orientations of graphs with a dominating clique
本文针对含支配团的图，证明了Hefetz等人提出的“每个连通图都存在反魔法定向”的猜想成立，完善了反魔法定向领域的相关结论。
This paper contains a description of a connection between the matching arrangement and the matching polyhedron. A bijection between regions of the matching arragement and LP-orientations of the matching polyhedron is constructed. This bijection allows to calculate the number of LP-orientations of the matching polyhedron with the characteristic polynomial of the matching arrangement.
Greene and Zaslavsky proved that the number of acyclic orientations of a graph with a unique sink is, up to sign, the linear coefficient of the chromatic polynomial. We give three new proofs of this result using pure induction, noncommutative symmetric functions, and an algorithmic bijection.
The orientation completion problem for a fixed class of oriented graphs asks whether a given partially oriented graph can be completed to an oriented graph in the class. Orientation completion problems have been studied recently for several classes of oriented graphs, yielding both polynomial time solutions and NP-completeness results. Local tournaments are a well-structured class of oriented graphs that generalize tournaments and their underlying graphs are intimately related to proper circular-arc graphs. Proper interval graphs are precisely those which can be oriented as acyclic local tourn…
We count orientations of $G(n,p)$ avoiding certain classes of oriented graphs. In particular, we study $T_r(n,p)$, the number of orientations of the binomial random graph $G(n,p)$ in which every copy of $K_r$ is transitive, and $S_r(n,p)$, the number of orientations of $G(n,p)$ containing no strongly connected copy of $K_r$. We give the correct order of growth of $\log T_r(n,p)$ and $\log S_r(n,p)$ up to polylogarithmic factors; for orientations with no cyclic triangle, this significantly improves a result of Allen, Kohayakawa, Mota and Parente. We also discuss the problem for a single forbidd…
We consider the class of simple graphs with large algebraic connectivity (the second-smallest eigenvalue of the Laplacian matrix). For this class of graphs we determine the asymptotic behavior of the number of Eulerian orientations. In addition, we establish some new properties of the Laplacian matrix, as well as an estimate of a conditionality of matrices with the asymptotic diagonal predominance
Subgraphs versus Orientations: Infinite families of equidistributions
本文探讨了子图与定向之间在连接性约束下的等分布关系，证明了特定连接性条件下的子图与定向数量相等，并扩展到考虑循环反转等等价类的计数问题。
Encoding and Enumerating Acyclic Orientations of Graphs
本文研究图的非循环定向编码与枚举，提出完全多部图的向量编码和递归构造方法，并给出若干闭公式及任意图非循环定向数目的精确上下界。
Let P a polytope and let G(P) be the graph of P. Following Gil Kalai, we say that an acyclic orientation O of G(P) is good if, for every non-empty face F of P, the induced graph G(F) has exactly one sink. Gil Kalai gave a simple way to tell a simple polytope from the good orientations of its graph. This article is a broader study of ``good orientations'' (of the graphs) on matroid polytopes.
Arc-distinguishing of orientations of graphs
本文研究图的定向对区分指数的影响，建立二分图和树的界与精确关系，并研究刚性定向，回答了关于不平衡完全二分图的开放问题。
Jaeger-type orientations of random regular graphs
研究随机正则图的p-定向，证明对于某些(d,p)值，图具有高概率的p-定向，包括Jaeger猜想中的p=3,4情况。
Counting orientations of random graphs with no directed k-cycles
该研究确定了二项随机图不含指定长度有向环的定向数的增长阶，解决了相关猜想，推进了随机图定向计数领域的研究。
Given a digraph $D$ with $m$ arcs and a bijection $τ: A(D)\rightarrow \{1, 2, \ldots, m\}$, we say $(D, τ)$ is an antimagic orientation of a graph $G$ if $D$ is an orientation of $G$ and no two vertices in $D$ have the same vertex-sum under $τ$, where the vertex-sum of a vertex $u$ in $D$ under $τ$ is the sum of labels of all arcs entering $u$ minus the sum of labels of all arcs leaving $u$. Hefetz, Mütze, and Schwartz in 2010 initiated the study of antimagic orientations of graphs, and conjectured that every connected graph admits an antimagic orientation. This conjecture seems hard, and few …
Building on previous work by Cameron et al. in [3], we give a recurrence for computing the number of acyclic orientations of complete $k$-partite graphs, which can be implemented to obtain a dynamic programming algorithm running in time $n^{O(k)}$, where $n$ is the number of vertices in the graph. We prove our result by using a relationship between the number of acyclic orientations and the number of Hamiltonian paths in complete $k$-partite graphs and providing a recurrence for the latter quantity. We give a simple extension of our algorithm to the situation when we are an edge removal away f…
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…
Let $G$ and $H$ be graphs, and $G\boxtimes H$ the strong product of $G$ and $H$. We prove that for any connected graphs $G$ and $H$ there is a strongly connected orientation $D$ of $G\boxtimes H$ such that ${\rm diam}(D)\leq 2r+15$, where $r$ is the radius of $G\boxtimes H$.This improves the general bound ${\rm diam}(D)\leq 2r^2+2r$ for arbitrary graphs, proved by Chv${\rm \acute{a}}$tal and Thomassen.
Given a digraph $D$ with $m $ arcs, a bijection $τ: A(D)\rightarrow \{1, 2, \ldots, m\}$ is an antimagic labeling of $D$ if no two vertices in $D$ have the same vertex-sum, where the vertex-sum of a vertex $u $ in $D$ under $τ$ is the sum of labels of all arcs entering $u$ minus the sum of labels of all arcs leaving $u$. We say $(D, τ)$ is an antimagic orientation of a graph $G$ if $D$ is an orientation of $G$ and $τ$ is an antimagic labeling of $D$. Motivated by the conjecture of Hartsfield and Ringel from 1990 on antimagic labelings of graphs, Hefetz, Mütze, and Schwartz in 2010 initiated th…
Orientations of graphs with at most one directed path between every pair of vertices
本文研究了允许每对顶点间至多有一条有向路径的KT定向图，构造了具有次线性独立数的图族，证明了判定该定向的NP完全性，并给出了最大度为3图的多项式算法。
Let $G$ be a connected graph and let $T$ be a spanning tree of $G$. A partial orientation $σ$ of $G$ respect to $T$ is an orientation of the edges of $G$ except those edges of $T$, the resulting graph associated with which is denoted by $G_T^σ$. In this paper we prove that there exists a partial orientation $σ$ of $G$ respect to $T$ such that the largest eigenvalue of the Hermitian adjacency matrix of $G_T^σ$ is at most the largest absolute value of the roots of the matching polynomial of $G$.
On orientations preserving edge-connectivity in infinite graphs
本文证明可数边端的2k-边连通图存在k-弧连通定向，推广了局部有限情形，并探讨了Nash-Williams猜想的拓扑路径替代方案，证明其对有限分离图成立。
A labeling of a digraph $D$ with $m$ arcs is a bijection from the set of arcs of $D$ to $\{1, \ldots, m\}$. A labeling of $D$ is antimagic if no two vertices in $D$ have the same vertex-sum, where the vertex-sum of a vertex $u\in V(D)$ for a labeling is the sum of labels of all arcs entering $u$ minus the sum of labels of all arcs leaving $u$. Motivated by the conjecture of Hartsfield and Ringel from 1990 on antimagic labelings of graphs, Hefetz, Mütze, and Schwartz [On antimagic directed graphs, J Graph Theory 64 (2010) 219--232] initiated the study of antimagic labelings of digraphs, and con…
r-orientation of a signed graph and its application on coronae of signed graphs
本文引入反向定向（r-orientation）概念定义了符号图的冠图变体，利用符号coronal获得了正则符号图在r-定向下多种矩阵的特征多项式，并借此构造了无限多对共谱符号图族。
Alternating odd cycles and orientations of Kneser-like graphs
本文研究特定图的定向问题，聚焦拓扑χ-染色图，探讨其是否存在避免交错奇环的定向或使所有最短奇环为交错奇环的定向，并针对Kneser图等多个图族给出结果。
A $labeling$ of a digraph $D$ with $m$ arcs is a bijection from the set of arcs of $D$ to $\{1,2,\ldots,m\}$. A labeling of $D$ is $antimagic$ if no two vertices in $D$ have the same vertex-sum, where the vertex-sum of a vertex $u \in V(D)$ for a labeling is the sum of labels of all arcs entering $u$ minus the sum of labels of all arcs leaving $u$. An antimagic orientation $D$ of a graph $G$ is $antimagic$ if $D$ has an antimagic labeling. Hefetz, M$\ddot{u}$tze and Schwartz in [J. Graph Theory 64(2010)219-232] raised the question: Does every graph admits an antimagic orientation? It had been …
Chains and unique transitive orientations of prime graphs
本文用链工具简洁证明素图至多有两个传递定向，且任意传递定向在翻转下唯一，为Gallai定理提供新证明。
The Wiener index of a strong digraph $D$ is defined as the sum of the distances between all ordered pairs of vertices. This definition has been extended to digraphs that are not necessarily strong by defining the distance from a vertex $a$ to a vertex $b$ as $0$ if there is no path from $a$ to $b$ in $D$. Knor, uSkrekovski and Tepeh [Some remarks on Wiener index of oriented graphs. Appl.\ Math.\ Comput.\ {\bf 273}] considered orientations of graphs with maximum Wiener index. The authors conjectured that for a given tree $T$, an orientation $D$ of $T$ of maximum Wiener index always contains a v…
Let $m\ge 1$ be an integer and $G$ be a graph with $m$ edges. We say that $G$ has an antimagic orientation if $G$ has an orientation $D$ and a bijection $τ:A(D)\rightarrow \{1,2,\cdots,m\}$ such that no two vertices in $D$ have the same vertex-sum under $τ$, where the vertex-sum of a vertex $u$ in $D$ under $τ$ is the sum of labels of all arcs entering $u$ minus the sum of labels of all arcs leaving $u$. Hefetz, Mütze and Schwartz [J. Graph Theory, 64: 219-232, 2010] conjectured that every connected graph admits an antimagic orientation. The conjecture was confirmed for certain classes of grap…
We present a new correspondence between acyclic orientations and coloring of a signed graph (symmetric graph). Goodall et al. introduced a bivariate chromatic polynomial $χ_G(k,l)$ that counts the number of signed colorings using colors $0,\pm1,\dots,\pm k$ along with $l-1$ symmetric colors $0_1,\dots,0_{l-1}$. We show that the evaluation of the bivariate chromatic polynomial $|χ_G(-1,2)|$ is equal to the number of acyclic orientations of the signed graph modulo the equivalence relation generated by swapping sources and sinks. We present three proofs of this fact, a proof using toric hyperplan…
Given a connected and bridgeless graph $G$, let $\mathscr{D}(G)$ be the family of strong orientations of $G$. The orientation number of $G$ is defined to be $\bar{d}(G):=min\{d(D)|D\in \mathscr{D}(G)\}$, where $d(D)$ is the diameter of the digraph $D$. In this paper, we focus on the orientation number of complete tripartite graphs. We prove a conjecture raised by Rajasekaran and Sampathkumar. Specifically, for $q\ge p\ge 3$, if $\bar{d}(K(2,p,q))=2$, then $q\le{{p}\choose{\lfloor{p/2}\rfloor}}$. We also present some sufficient conditions on $p$ and $q$ for $\bar{d}(K(p,p,q))=2$.
Gessel and Sagan investigated the Tutte polynomial, $T(x,y)$ using depth first search, and applied their techniques to show that the number of acyclic partial orientations of a graph is $2^gT(3,1/2)$. We provide a short deletion-contraction proof of this result and demonstrate that dually, the number of strongly connected partial orientations is $2^{n-1}T(1/2,3)$. We then prove that the number of partial orientations modulo cycle reversals is $2^gT(3,1)$ and the number of partial orientations modulo cut reversals is $2^{n-1}T(1,3)$. To prove these results, we introduce cut and cycle minimal pa…
The orientation completion problem for a class of oriented graphs asks whether a given partially oriented graph can be completed to an oriented graph in the class by orienting the unoriented edges of the partially oriented graph. Orientation completion problems have been studied recently for several classes of oriented graphs, yielding both polynomial time solutions as well as NP-completeness results. Local tournaments are a well-structured class of oriented graphs that generalize tournaments and their underlying graphs are intimately related to proper circular-arc graphs. According to Skrien,…
Seymour's Second Neighborhood Conjecture (SNC) states that every oriented graph contains a vertex whose second neighborhood is as large as its first neighborhood. We investigate the SNC for orientations of both binomial and pseudo random graphs, verifying the SNC asymptotically almost surely (a.a.s.) (i) for all orientations of $G(n,p)$ if $\limsup_{n\to\infty} p < 1/4$; and (ii) for a uniformly-random orientation of each weakly $(p,A\sqrt{np})$-bijumbled graph of order $n$ and density $p$, where $p=Ω(n^{-1/2})$ and $1-p = Ω(n^{-1/6})$ and $A>0$ is a universal constant independent of both $n$ …
In this paper we study graphs which admit acyclic orientations that contain a pair of arc-disjoint out-branching and in-branching (such an orientation is called good) and we focus on edge-minimal such graphs. A 2T-graph is a graph whose edge set can be decomposed into two edge-disjoint spanning trees. Vertex-minimal 2T-graphs with at least two vertices which are known as generic circuits play an important role in rigidity theory for graphs. We prove that every generic circuit has a good orientation. Using this result we prove that if $G$ is 2T-graph whose vertex set has a partition $V_1,V_2,\l…
A bull is a graph with five vertices $r, y, x, z, s$ and five edges $ry$, $yx$, $yz$, $xz$, $zs$. A graph $G$ is bull-reducible if no vertex of $G$ lies in two bulls. We prove that every bull-reducible Berge graph $G$ that contains no antihole is weakly chordal, or has a homogeneous set, or is transitively orientable. This yields a fast polynomial time algorithm to color exactly the vertices of such a graph.
This is the first one of a series of papers on association of orientations, lattice polytopes, and abelian group arrangements to graphs. The purpose is to interpret the integral and modular tension polynomials of graphs at zero and negative integers. The whole exposition is put under the framework of subgroup arrangements and the application of Ehrhart polynomials. Such viewpoint leads to the following main results of the paper: (i) the reciprocity law for integral tension polynomials; (ii) the reciprocity law for modular tension polynomials; and (iii) a new interpretation for the value of the…
Given a graph $G$ with only even degrees let $\varepsilon(G)$ denote the number of Eulerian orientations, and let $h(G)$ denote the number of half graphs, that is, subgraphs $F$ such that $d_F(v)=d_G(v)/2$ for each vertex $v$. Recently, Borbényi and Csikvári proved that $\varepsilon(G)\geq h(G)$ holds true for all Eulerian graphs with equality if and and only if $G$ is bipartite. In this paper we give a simple new proof of this fact, and we give identities and inequalities for the number of Eulerian orientations and half graphs of a $2$-cover of a graph $G$.
The main contribution of this paper is a formula for the number of acyclic orientations of a complete bipartite, $K_{n_1,n_2},$ revealing that it is equal to the poly-Bernoulli number $B_{n_1}^{(-n_2)}$ introduced in 1997 by Kaneko. We also give a simple bijective identification of acyclic orientations and lonesum matrices, confirming a 2008 result of Brewbaker, and show that the poly-Bernoulli numbers behave concavely, as well being symmetric. A second goal is to explore the behaviour of more general complete $r$-partite graphs in the space of the number of acyclic orientations. We prove that…
A semi-proper orientation of a given graph $G$, denoted by $(D,w)$, is an orientation $D$ with a weight function $w: A(D)\rightarrow \mathbb{Z}_+$, such that the in-weight of any adjacent vertices are distinct, where the in-weight of $v$ in $D$, denoted by $w^-_D(v)$, is the sum of the weights of arcs towards $v$. The semi-proper orientation number of a graph $G$, denoted by $\overrightarrowχ_s(G)$, is the minimum of maximum in-weight of $v$ in $D$ over all semi-proper orientation $(D,w)$ of $G$. This parameter was first introduced by Dehghan (2019). When the weights of all edges eqaul to one,…
A theory of orientation on gain graphs (voltage graphs) is developed to generalize the notion of orientation on graphs and signed graphs. Using this orientation scheme, the line graph of a gain graph is studied. For a particular family of gain graphs with complex units, matrix properties are established. As with graphs and signed graphs, there is a relationship between the incidence matrix of a complex unit gain graph and the adjacency matrix of the line graph.
Characterization of graphs without even $F$-orientations
本文刻画了无偶数$F$-定向的$1$-可扩展图，特别对连通度至少4的图和$k$-正则图（$k \geq 3$）给出了完整分类。
Let $G$ be a graph embedded on an orientable surface. Given a class ${\cal C}$ of facial circuits of $G$ as a forbidden class, we give a sufficient-necessary condition for that an $α$-orientation (orientation with prescribed out-degrees) of $G$ can be transformed into another by a sequence of flips on non-forbidden circuits and further give an explicit formula for the minimum number of such flips. We also consider the connection among all $α$-orientations by defining a directed graph ${\bf D}({\cal C})$, namely the ${\cal C}$-forbidden flip graph. We show that if ${\cal C}\not=\emptyset$, then…
Given an underlying undirected simple graph, we consider the set of all acyclic orientations of its edges. Each of these orientations induces a partial order on the vertices of our graph and, therefore, we can count the number of linear extensions of these posets. We want to know which choice of orientation maximizes the number of linear extensions of the corresponding poset, and this problem will be solved essentially for comparability graphs and odd cycles, presenting several proofs. The corresponding enumeration problem for arbitrary simple graphs will be studied, including the case of rand…
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