Orientations Of Graphs Research Articles

The Chromatic Symmetric Function of a Graph Centred at a Vertex

We discover new linear relations between the chromatic symmetric functions of certain sequences of graphs and apply these relations to find new families of $e$-positive unit interval graphs. Motivated by the results of Gebhard and Sagan, we revisit their ideas and reinterpret their equivalence relation in terms of a new quotient algebra of NCSym. We investigate the projection of the chromatic symmetric function $Y_G$ in noncommuting variables in this quotient algebra, which defines $y_{G : v}$, the chromatic symmetric function of a graph $G$ centred at a vertex $v$. We then apply our methods to $y_{G : v}$ and find new families of unit interval graphs that are $(e)$-positive, a stronger condition than classical $e$-positivity, thus confirming new cases of the $(3+1)$-free conjecture of Stanley and Stembridge. In our study of $y_{G : v}$, we also describe methods of constructing new $e$-positive graphs from given $(e)$-positive graphs and classify the $(e)$-positivity of trees and cut vertices. We moreover construct a related quotient algebra of NCQSym to prove theorems relating the coefficients of $y_{G : v}$ to acyclic orientations of graphs, including a noncommutative refinement of Stanley's sink theorem.

Directed graphs with lower orientation Ramsey thresholds

We investigate the threshold p⃗H = p⃗H (n) for the Ramsey-type property G(n, p) → ⃗H, where G(n, p) is the binomial random graph and G → ⃗H indicates that every orientation of the graph G contains the oriented graph ⃗ H as a subdigraph. Similarly to the classical Ramsey setting, the upper bound p⃗H ⩽ Cn−1/m2(⃗ H) is known to hold for some constant C = C( ⃗ H), where m2(⃗H) denotes the maximum 2-density of the underlying graph H of ⃗ H. While this upper bound is indeed the threshold for some ⃗H, this is not always the case. We obtain examples arising from rooted products of orientations of sparse graphs (such as forests, cycles and, more generally, subcubic {K3, K3,3}-free graphs) and arbitrarily rooted transitive triangles.

Interval colourable orientations of graphs

A graph is interval colourable if it admits a proper edge colouring in which for each vertex the set of colours of edges that are incident with the vertex is an interval of integers. An oriented graph is interval colourable if it admits a proper arc colouring in which for each vertex the set of colours of in-arcs and the set of colours of out-arcs that are incident with the vertex are both intervals of integers. In this paper we apply a classical approach and a new special trail technique to confirm the existence of interval colourable orientations of graphs from three classes: i) all k-trees with Δ(G)≤2k and k∈{2,3,4} and all k-paths with arbitrary k∈N; ii) graphs in which the length of every even closed trail is at least 2s and each such a trail has common edges with no more than d other even closed trails and 21−2se(d+1)≤1; iii) graphs that are decomposable into a bipartite interval colourable graph and a graph whose each connected component has at most one cycle that, if it exists, has odd length. Consequently, cactus graphs, graphs that are homeomorphic to Halin graphs, full subdivision graphs and others have interval colourable orientations.

Arc Connectivity and Submodular Flows in Digraphs

Let D=(V,A)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$D=(V,A)$$\\end{document} be a digraph. For an integer k≥1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$k\\ge 1$$\\end{document}, a k-arc-connected flip is an arc subset of D such that after reversing the arcs in it the digraph becomes (strongly) k-arc-connected. The first main result of this paper introduces a sufficient condition for the existence of a k-arc-connected flip that is also a submodular flow for a crossing submodular function. More specifically, given some integer τ≥1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ au \\ge 1$$\\end{document}, suppose dA+(U)+(τk-1)dA-(U)≥τ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$d_A^+(U)+(\\frac{\ au }{k}-1)d_A^-(U)\\ge \ au $$\\end{document} for all U⊊V,U≠∅\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$U\\subsetneq V, U\ e \\emptyset $$\\end{document}, where dA+(U)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$d_A^+(U)$$\\end{document} and dA-(U)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$d_A^-(U)$$\\end{document} denote the number of arcs in A leaving and entering U, respectively. Let C\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\mathcal {C}}$$\\end{document} be a crossing family over ground set V, and let f:C→Z\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$f:{\\mathcal {C}}\\rightarrow {\\mathbb {Z}}$$\\end{document} be a crossing submodular function such that f(U)≥kτ(dA+(U)-dA-(U))\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$f(U)\\ge \\frac{k}{\ au }(d_A^+(U)-d_A^-(U))$$\\end{document} for all U∈C\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$U\\in {\\mathcal {C}}$$\\end{document}. Then D has a k-arc-connected flip J such that f(U)≥dJ+(U)-dJ-(U)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$f(U)\\ge d_J^+(U)-d_J^-(U)$$\\end{document} for all U∈C\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$U\\in {\\mathcal {C}}$$\\end{document}. The result has several applications to Graph Orientations and Combinatorial Optimization. In particular, it strengthens Nash-Williams’ so-called weak orientation theorem, and proves a weaker variant of Woodall’s conjecture on digraphs whose underlying undirected graph is τ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ au $$\\end{document}-edge-connected. The second main result of this paper is even more general. It introduces a sufficient condition for the existence of capacitated integral solutions to the intersection of two submodular flow systems. This sufficient condition implies the classic result of Edmonds and Giles on the box-total dual integrality of a submodular flow system. It also has the consequence that in a weakly connected digraph, the intersection of two submodular flow systems is totally dual integral.

Partially broken orientations of Eulerian graphs on closed surfaces

In this paper, we discuss orientations given to edges of Eulerian graphs embedded on closed surfaces, and give a characterization of such a graph having a good orientation, i.e., incoming edges and outgoing edges appear alternately around each vertex of the graph. Furthermore, we consider the partially broken orientations of Eulerian graphs embedded on closed surfaces, that is, one in which only for each of specified vertices, incoming edges and outgoing edges appear alternately around it. We discuss a little about good orientations defined for Eulerian graphs drawn on closed surfaces with crossings, and for general connected graphs, i.e., not necessarily Eulerian graphs, embedded on closed surfaces.

Orientations of graphs with maximum Wiener index

AbstractIn this paper, we study the Wiener index of the orientation of trees and theta‐graphs. An orientation of a tree is called no‐zig‐zag if there is no subpath in which edges change the orientation twice. Knor, Škrekovski, and Tepeh conjectured that every orientation of a tree achieving the maximum Wiener index is no‐zig‐zag. We disprove this conjecture by constructing a counterexample. Knor, Škrekovski, and Tepeh conjectured that among all orientations of the theta‐graph with and , the maximum Wiener index is achieved by the one in which the union of the paths between and forms a directed cycle of length , where and are the vertex of degree 3. We confirm the validity of the conjecture.

Counting orientations of random graphs with no directed k‐cycles

AbstractFor every , we determine the order of growth, up to polylogarithmic factors, of the number of orientations of the binomial random graph containing no directed cycle of length . This solves a conjecture of Kohayakawa, Morris and the last two authors.

On Seymour's and Sullivan's second neighbourhood conjectures

AbstractFor a vertex of a digraph, (, respectively) is the number of vertices at distance 1 from (to, respectively) and is the number of vertices at distance 2 from . In 1995, Seymour conjectured that for any oriented graph there exists a vertex such that . In 2006, Sullivan conjectured that there exists a vertex in such that . We give a sufficient condition in terms of the number of transitive triangles for an oriented graph to satisfy Sullivan's conjecture. In particular, this implies that Sullivan's conjecture holds for all orientations of planar graphs and triangle‐free graphs. An oriented graph is an oriented split graph if the vertices of can be partitioned into vertex sets and such that is an independent set and induces a tournament. We also show that the two conjectures hold for some families of oriented split graphs, in particular, when induces a regular or an almost regular tournament.

On semi-transitive orientability of split graphs

A directed graph is semi-transitive if and only if it is acyclic and for any directed path u1→u2→⋯→ut, t≥2, either there is no edge from u1 to ut or all edges ui→uj exist for 1≤i<j≤t. An undirected graph is semi-transitive if it admits a semi-transitive orientation. Recognizing semi-transitive orientability of a graph is an NP-complete problem.A split graph is a graph in which the vertices can be partitioned into a clique and an independent set. Semi-transitive orientability of split graphs was recently studied in a series of papers in the literature. The main result in this paper is proving that recognition of semi-transitive orientability of split graphs can be done in a polynomial time.

On the Wiener index of orientations of graphs

On the Wiener index of orientations of graphs

Factor-of-iid balanced orientation of non-amenable graphs

We show that if a non-amenable, quasi-transitive, unimodular graph G has all degrees even then it has a factor-of-iid balanced orientation, meaning each vertex has equal in- and outdegree. This result involves extending earlier spectral-theoretic results on Bernoulli shifts to the Bernoulli graphings of quasi-transitive, unimodular graphs.As a consequence, we also obtain that when G is regular (of either odd or even degree) and bipartite, it has a factor-of-iid perfect matching. This generalizes a result of Lyons and Nazarov beyond transitive graphs.

Seymour's Second Neighborhood Conjecture for orientations of (pseudo)random graphs

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.)1.for all orientations of G(n,p) if limsupn→∞p<1/4; and2.for a uniformly-random orientation of each weakly (p,Anp)-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 and p. We also show that a.a.s. the SNC holds for almost every orientation of G(n,p). More specifically, we prove that a.a.s.1.for all ε>0 and p=p(n) with limsupn→∞p≤2/3−ε, every orientation of G(n,p) with minimum outdegree Ωε(n) satisfies the SNC; and2.for all p=p(n), a random orientation of G(n,p) satisfies the SNC. We remark that either (iii) or (iv) confirms the SNC for almost every oriented graph.

Fixed-parameter algorithms for graph constraint logic

Non-deterministic constraint logic (NCL) is a simple model of computation based on orientations of a constraint graph with edge weights and vertex demands. NCL captures PSPACE and has been a useful tool for proving algorithmic hardness of many puzzles, games, and reconfiguration problems. In particular, its usefulness stems from the fact that it remains PSPACE-complete even under severe restrictions of the weights (e.g., only edge-weights one and two are needed) and the structure of the constraint graph (e.g., planar and/or graphs of bounded bandwidth). While such restrictions on the structure of constraint graphs do not seem to limit the expressiveness of NCL, the building blocks of the constraint graphs cannot be limited without losing expressiveness: We consider as parameters the number of weight-one edges and the number of weight-two edges of a constraint graph, as well as the number of and or or vertices of an and/or constraint graph. We show that NCL is fixed-parameter tractable (FPT) for any of these parameters. In particular, for NCL parameterized by the number of weight-one edges or the number of and vertices, we obtain a linear kernel. It follows that, in a sense, NCL as introduced by Hearn and Demaine is defined in the most economical way for the purpose of capturing PSPACE.

Using the minimum and maximum degrees to bound the diameter of orientations of bridgeless graphs

Using the minimum and maximum degrees to bound the diameter of orientations of bridgeless graphs

Monotone Edge Flips to an Orientation of Maximum Edge-Connectivity à la Nash-Williams

We initiate the study of k -edge-connected orientations of undirected graphs through edge flips for k ≥ 2. We prove that in every orientation of an undirected 2k -edge-connected graph, there exists a sequence of edges such that flipping their directions one by one does not decrease the edge connectivity, and the final orientation is k -edge connected. This yields an “edge-flip based” new proof of Nash-Williams’ theorem: A undirected graph G has a k -edge-connected orientation if and only if G is 2k -edge connected. As another consequence of the theorem, we prove that the edge-flip graph of k -edge-connected orientations of an undirected graph G is connected if G is (2k+2) -edge connected. This has been known to be true only when k=1 .

$st$-Orientations with Few Transitive Edges

The problem of orienting the edges of an undirected graph such that the resulting digraph is acyclic and has a single source $s$ and a single sink $t$ has a long tradition in graph theory and is central to many graph drawing algorithms. Such an orientation is called an $st$-orientation. We address the problem of computing $st$-orientations of undirected graphs with the minimum number of transitive edges. We prove that the problem is NP-hard in the general case. For planar graphs we describe an ILP (Integer Linear Programming) model that is fast in practice, namely it takes on average less than 1 second for graphs with up to 100 vertices, and about 10 seconds for larger instances with up to 1000 vertices. We experimentally show that optimum solutions significantly reduce (35% on average) the number of transitive edges with respect to unconstrained $st$-orientations computed via classical $st$-numbering algorithms. Moreover, focusing on popular graph drawing algorithms that apply an $st$-orientation as a preliminary step, we show that reducing the number of transitive edges leads to drawings that are much more compact (with an improvement between 30% and 50% for most of the instances).

An analogue of quasi-transitivity for edge-coloured graphs

We extend the notion of quasi-transitive orientations of graphs to 2-edge-coloured graphs. By relating quasi-transitive $2$-edge-colourings to an equivalence relation on the edge set of a graph, we classify those graphs that admit a quasi-transitive $2$-edge-colouring. As a contrast to Ghouila-Houri's classification of quasi-transitively orientable graphs as comparability graphs, we find quasi-transitively $2$-edge-colourable graphs do not admit a forbiddden subgraph characterization. Restricting the problem to comparability graphs, we show that the family of uniquely quasi-transitively orientable comparability graphs is exactly the family of comparabilty graphs that admit no quasi-transitive $2$-edge-colouring.

Аттракторы и циклические состояния в конечных динамических системах ориентаций полных графов

Graph models occupy an important place in information security tasks. Finite dynamic systems of complete graphs orientations are considered. States of a dynamic system (ΓKn , α), n ⩾ 1, are all possible orientations of a given complete graph Kn, and the evolutionary function transforms the complete graph orientation by reversing all arcs that go into sinks, and there are no other differences between the given and the next digraphs. The cyclic (belonging to attractors) states of the system are characterized, namely, state belongs to an attractor, if and only if it hasn’t a sink or its indegrees vector is (n − 1, n − 2, . . . , 0). A table is given with the number of cyclic and non-cyclic states in the systems of complete graphs orientations with the number of vertices from 1 to 8 inclusive. The formation of attractors of the system, their type and length are described, namely, there are attractors of length 1, each of which is formed by state without sink, and attractors of length n, each of which is formed by such states −→G ∈ ΓKn , in which indegrees vector is (n − 1, n − 2, . . . , 0), wherein each such attractor represents a circuit, in which each next state is obtained from the previous one as follows: if −→G has vector of indegrees of its vertices in the order of their enumeration (d −(v1), d−(v2), . . . , d−(vn)), then α( −→G) ∈ ΓKn has vector of indegrees of its vertices in the order of their enumeration (d −(v1) + 1, d−(v2) + 1, . . . , d−(vn) + 1), where the addition is calculated modulo n, and only these attractors. Note that in the considered finite dynamic systems the number of attractors of length n is equal to (n − 1)! and the number belonging to the attractors states is equal to n!. A table is given with the corresponding number of attractors in the systems of complete graphs orientations with the number of vertices from 1 to 8 inclusive.

Flips on homologous orientations of surface graphs with prescribed forbidden facial cycles

Let G be a graph embedded on an orientable surface. Given a class C of facial cycles of G as a forbidden class, we give a sufficient and necessary condition such that an α-orientation (orientation with prescribed out-degrees) of G can be transformed into another by a sequence of flips on non-forbidden cycles and further give an explicit formula for the minimum number of such flips.

Weighted Modulo Orientations of Graphs and Signed Graphs

Given a graph $G$ and an odd prime $p$, for a mapping $f: E(G) \to {\mathbb Z}_p\setminus\{0\}$ and a ${\mathbb Z}_p$-boundary $b$ of $G$, an orientation $\tau$ is called an $(f,b;p)$-orientation if the net out $f$-flow is the same as $b(v)$ in ${\mathbb Z}_p$ at each vertex $v\in V(G)$ under orientation $D$. This concept was introduced by Esperet et al. (2018), generalizing mod $p$-orientations and closely related to Tutte's nowhere zero 3-flow conjecture. They proved that $(6p^2 - 14p + 8)$-edge-connected graphs have all possible $(f,b;p)$-orientations. In this paper, the framework of such orientations is extended to signed graph through additive bases. We also study the $(f,b;p)$-orientation problem for some (signed) graphs families including complete graphs, chordal graphs, series-parallel graphs and bipartite graphs, indicating that much lower edge-connectivity bound still guarantees the existence of such orientations for those graph families.