The set A of distinct scores of the vertices of an oriented bipartite graph D(U, V) is called its score set. We consider the following question: given a finite, nonempty set A of positive integers, is there an oriented bipartite graph D(U, V) such that score set of D(U, V) is A? We conjecture that there is an affirmative answer, and verify this conjecture when $\mid A\mid $ = 1, 2, 3, or when A is a geometric or arithmetic progression.
Given a graph $G$, let $G^σ$ be an oriented graph of $G$ with the orientation $σ$ and skew-adjacency matrix $S(G^σ)$. Then the spectrum of $S(G^σ)$ is called the skew-spectrum of $G^σ$, denoted by $Sp_S(G^σ)$. It is known that a graph $G$ is bipartite if and only if there is an orientation $σ$ of $G$ such that $Sp_S(G^σ)=iSp(G)$. In [D. Cui, Y. Hou, On the skew spectra of Cartesian products of graphs, Electron. J. Combin. 20(2013), #P19], Cui and Hou conjectured that such orientation of a bipartite graph is unique under switching-equivalence. In this paper, we prove that the conjecture is true…
In this paper, we study oriented bipartite graphs. In particular, we introduce "bitransitive" graphs. Several characterizations of bitransitive bitournaments are obtained. We show that bitransitive bitounaments are equivalent to acyclic bitournaments. As applications, we characterize acyclic bitournaments with Hamiltonian paths, determine number of non-isomorphic acyclic bitournaments of a given order, and solve the graph-isomorphism problem in linear time for acyclic bitournaments. Next, we prove the well-known Caccetta-H$\ddot{\textrm{a}}$ggkvist Conjecture for oriented bipartite graphs for …
Given non-negative integers m, n, h and k with $ m\geq h>1 $ and $ n\geq k>1, $an [h-k]-bipartite hypertournament on $ m+n$ vertices is a triple $(U,V,A) $, where U and V are two sets of vertices with $| U| =m$ and $ | V| =n,$ and $A$ is a set of $(h+k) -$ tuples of vertices, called arcs, with exactly $h$ vertices from $U$ and exactly $k$ vertices from $V$, such that any $h+k$ subsets $ U_{1}\cup V_{1}$ of $U\cup V, A$ contains exactly one of the $(h+k) ! (h+k) -$tuples whose entries belong to $U_{1}\cup V_{1}.$ We obtain necessary and sufficient conditions for a pair of non-decreasing sequenc…
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…
The permanental polynomial of a graph $G$ is $π(G,x)\triangleq\mathrm{per}(xI-A(G))$. From the result that a bipartite graph $G$ admits an orientation $G^e$ such that every cycle is oddly oriented if and only if it contains no even subdivision of $K_{2,3}$, Yan and Zhang showed that the permanental polynomial of such a bipartite graph $G$ can be expressed as the characteristic polynomial of the skew adjacency matrix $A(G^e)$. In this paper we first prove that this equality holds only if the bipartite graph $G$ contains no even subdivision of $K_{2,3}$. Then we prove that such bipartite graphs …
It is well known that a plane graph is Eulerian if and only if its geometric dual is bipartite. We extend this result to partial duals of plane graphs. We then characterize all bipartite partial duals of a plane graph in terms of oriented circuits in its medial graph.
In this paper we obtain a new sufficient condition for the existence of directed cycles of length 4 in oriented bipartite graphs. As a corollary, a conjecture of H. Li is confirmed. As an application, a sufficient condition for the existence of rainbow cycles of length 4 in bipartite edge-colored graphs is obtained.
In this work, we establish an analogue result of the Erdös-Stone theorem of weighted digraphs using Regularity Lemma of digraphs. We give a stability result of oriented graphs and digraphs with forbidden blow-up transitive triangle and show that almost all oriented graphs and almost all digraphs with forbidden blow-up transitive triangle are almost bipartite respectively.
The game INFLUENCE is a scoring combinatorial game that has been introduced in 2020 by Duchene et al. It is a good representative of Milnor's universe of scoring games, i.e. games where it is never interesting for a player to miss his turn. New general results are first given for this universe, by transposing the notions of mean and temperature derived from non-scoring combinatorial games. Such results are then applied to INFLUENCE to refine the case of unions of segments. The computational complexity of the score of the game is also solved and proved to be PSPACE-complete. We finally focus on…
Given a graph $G$, let $G^σ$ be an oriented graph of $G$ with the orientation $σ$ and skew-adjacency matrix $S(G^σ)$. Then the spectrum of $S(G^σ)$ consisting of all the eigenvalues of $S(G^σ)$ is called the skew-spectrum of $G^σ$, denoted by $Sp(G^σ)$. The skew energy of the oriented graph $G^σ$, denoted by $\mathcal{E}_S(G^σ)$, is defined as the sum of the norms of all the eigenvalues of $S(G^σ)$. In this paper, we give orientations of the Kronecker product $H\otimes G$ and the strong product $H\ast G$ of $H$ and $G$ where $H$ is a bipartite graph and $G$ is an arbitrary graph. Then we deter…
In 1981 Jackson showed that the diregular bipartite tournament (a complete bipartite graph whose edges are oriented so that every vertex has the same in- and outdegree) contains a Hamilton cycle, and conjectured that in fact the edge set of it can be partitioned into Hamilton cycles. We prove an approximate version of this conjecture: For every $c>1/2$ and $\varepsilon>0$ there exists $n_0$ such that every $cn$-regular bipartite digraph on $2n\geq n_0$ vertices contains $(1-\varepsilon)cn$ edge-disjoint Hamilton cycles.
Given an Eulerian digraph, we consider the genus distribution of its face-oriented embeddings. We prove that such distribution is log-concave for two families of Eulerian digraphs, thus giving a positive answer for these families to a question asked in Bonnington, Conder, Morton and McKenna (2002). Our proof uses real-rooted polynomials and the representation theory of the symmetric group $\mathbb{S}_n$. The result is also extended to some factorizations of the identity in $\mathbb{S}_n$ that are rotation systems of some families of one-face constellations.
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…
We study the family of graphs whose number of primitive cycles equals its cycle rank. It is shown that this family is precisely the family of ring graphs. Then we study the complete intersection property of toric ideals of bipartite graphs and oriented graphs. An interesting application is that complete intersection toric ideals of bipartite graphs correspond to ring graphs and that these ideals are minimally generated by Groebner bases. We prove that any graph can be oriented such that its toric ideal is a complete intersection with a universal Groebner basis determined by the cycles. It turn…
We study the Seifert surfaces of a link by relating the embeddings of graphs by using induced graphs. As applications, we prove that every link $L$ is the boundary of an oriented surface which is obtained from a graph embedding of a complete bipartite graph $K_{2,n}$, where all voltage assignments on the edges of $K_{2,n}$ are 0. We also provide an algorithm to construct such a graph diagram of a given link and demonstrate the algorithm by dealing with the links $4_1^2$ and $5_2$.
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…
A distinguishing r-vertex-labelling (resp. r-edge-labelling) of an undirected graph G is a mapping $λ$ from the set of vertices (resp. the set of edges) of G to the set of labels {1,. .. , r} such that no non-trivial automorphism of G preserves all the vertex (resp. edge) labels. The distinguishing number D(G) and the distinguishing index D (G) of G are then the smallest r for which G admits a distinguishing r-vertex-labelling or r-edge-labelling, respectively. The distinguishing chromatic number D $χ$ (G) and the distinguishing chromatic index D $χ$ (G) are defined similarly, with the additio…
In this short note, we extend the result of Galluccio, Goddyn, and Hell, which states that graphs of large girth excluding a minor are nearly bipartite. We also prove a similar result for the oriented chromatic number, from which follows in particular that graphs of large girth excluding a minor have oriented chromatic number at most $5$, and for the $p$th chromatic number $χ_p$, from which follows in particular that graphs $G$ of large girth excluding a minor have $χ_p(G)\leq p+2$.
A directed dominating set in a directed graph $D$ is a set $S$ of vertices of $V$ such that every vertex $u \in V(D) \setminus S$ has an adjacent vertex $v$ in $S$ with $v$ directed to $u$. The directed domination number of $D$, denoted by $γ(D)$, is the minimum cardinality of a directed dominating set in $D$. The directed domination number of a graph $G$, denoted $Γ_d(G)$, which is the maximum directed domination number $γ(D)$ over all orientations $D$ of $G$. The directed domination number of a complete graph was first studied by Erdös [Math. Gaz. 47 (1963), 220--222], albeit in disguised fo…
Let $D$ be a connected oriented graph. A set $S \subseteq V(D)$ is convex in $D$ if, for every pair of vertices $x, y \in S$, the vertex set of every $xy$-geodesic, ($xy$ shortest directed path) and every $yx$-geodesic in $D$ is contained in $S$. The convexity number, ${\rm con}(D)$, of a non-trivial oriented graph, $D$, is the maximum cardinality of a proper convex set of $D$. The strong convexity spectrum of the graph $G$, $S_{SC} (G)$, is the set $\{{ \rm con}(D) \colon\ D {\rm \ is \ a \ strong \ orientation \ of \ } G \}$. In this paper we prove that the problem of determining the convexi…
We study the eternal dominating number and the m-eternal dominating number on digraphs. We generalize known results on graphs to digraphs. We also consider the problem "oriented (m-)eternal domination", consisting in finding an orientation of a graph that minimizes its eternal dominating number. We prove that computing the oriented eternal dominating number is NP-hard and characterize the graphs for which the oriented m-eternal dominating number is 2. We also study these two parameters on trees, cycles, complete graphs, complete bipartite graphs, trivially perfect graphs and different kinds of…
A directed graph is oriented if it can be obtained by orienting the edges of a simple, undirected graph. For an oriented graph $G$, let $β(G)$ denote the size of a minimum feedback arc set, a smallest subset of edges whose deletion leaves an acyclic subgraph. A simple consequence of a result of Berger and Shor is that any oriented graph $G$ with $m$ edges satisfies $β(G) = m/2 - Ω(m^{3/4})$. We observe that if an oriented graph $G$ has a fixed forbidden subgraph $B$, the upper bound of $β(G) = m/2 - Ω(m^{3/4})$ is best possible as a function of the number of edges if $B$ is not bipartite, but …
Let $D$ be an orientation of a simple graph. Given $u,v\in V(D)$, a directed shortest $(u,v)$-path is a $(u,v)$-geodesic. $S \subseteq V(D)$ is convex if, for every $u,v \in S$, the vertices in each $(u,v)$-geodesic and in each $(v,u)$-geodesic are in $S$. For each $S \subseteq V(D)$ the (convex) hull of $S$, denoted by $[S]$, is the smallest convex set containing $S$. $S \subseteq V(D)$ is a hull set if $[S] = V(D)$. $S \subseteq V(D)$ is a geodetic set of $D$ if each vertex of $D$ lies in a $(u,v)$-geodesic, for some $u,v \in S$. The cardinality of a minimum hull set (resp. geodetic set) of …
Using the description of hypermaps with matchings, Goulden and Jackson have given an expression of the generating series of rooted bipartite maps in terms of the zonal polynomials. We generalize this approach to the case of constellations on non-oriented surfaces that have recently been introduced by Chapuy and Dołęga. A key step in the proof is an encoding of constellations with tuples of matchings. We consider a one parameter deformation of the generating series of constellations using Jack polynomials and we introduce the coefficients $c^λ_{μ^0,...,μ^k}(b)$ obtained by the expansion of thes…
We discuss conjectures on Hamiltonicity in cubic graphs (Tait, Barnette, Tutte), on the dichromatic number of planar oriented graphs (Neumann-Lara), and on even graphs in digraphs whose contraction is strongly connected (Hochstättler). We show that all of them fit into the same framework related to cuts in matchings. This allows us to find a counterexample to the conjecture of Hochstättler and show that the conjecture of Neumann-Lara holds for all planar graphs on at most 26 vertices. Finally, we state a new conjecture on bipartite cubic oriented graphs, that naturally arises in this setting.
A decorated surface S is an oriented surface with punctures and a finite set of marked points on the boundary, such that each boundary component has a marked point. We introduce ideal bipartite graphs on S. Each of them is related to a group G of type A, and gives rise to cluster coordinate systems on certain spaces of G-local systems on S. These coordinate systems generalize the ones assigned to ideal triangulations of S. A bipartite graph on S gives rise to a quiver with a canonical potential. The latter determines a triangulated 3d CY category with a cluster collection of spherical objects.…
