Given a graph $G$, its orientation covering number $σ(G)$ is the smallest non-negative integer $k$ with the property that we can choose $k$ orientations of $G$ such that whenever $x, y, z$ are vertices of $G$ with $xy,xz\in E(G)$ then there is a chosen orientation in which both $xy$ and $xz$ are oriented away from $x$. Esperet, Gimbel and King showed that $σ(G)\leq σ\left(K_{χ(G)}\right)$, where $χ(G)$ is the chromatic number of $G$, and asked whether we always have equality. In this note we prove that it is indeed always the case that $σ(G)=σ(K_{χ(G)})$. We also determine the exact value of $σ(K_n)$ explicitly for `most' values of $n$.
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 undirected biclique $K_{a,b}$ is a graph with vertices partitioned into two sets: a set $A$ containing $a$ vertices and a set $B$ containing $b$ vertices such that every vertex in set $A$ is connected to every vertex in set $B$, and such that no two vertices in the same set have an edge between them. A well-known result is that a minimum of $\lceil \log_2{n} \rceil$ bicliques graphs of any size are needed to edge-cover the complete graph on $n$ vertices. We prove a lower bound on minimum vertex-weighted biclique coverings of the complete graph $n$, and use this to prove an asymptotic formul…
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,…
We consider two possible extensions of a theorem of Thomassen characterizing the graphs admitting a 2-vertex-connected orientation. First, we show that the problem of deciding whether a mixed graph has a 2-vertex-connected orientation is NP-hard. This answers a question of Bang-Jensen, Huang and Zhu. For the second part, we call a directed graph $D=(V,A)$ $2T$-connected for some $T \subseteq V$ if $D$ is 2-arc-connected and $D-v$ is strongly connected for all $v \in T$. We deduce a characterization of the graphs admitting a $2T$-connected orientation from the theorem of Thomassen.
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.
We prove that in every cover of a Young diagram with $\binom{2k}{k}$ steps with generalized rectangles there is a row or a column in the diagram that is used by at least $k+1$ rectangles. We show that this is best-possible by partitioning any Young diagram with $\binom{2k}{k}-1$ steps into actual rectangles, each row and each column used by at most $k$ rectangles. This answers two questions by Kim et al. (2018). Our results can be rephrased in terms of local covering numbers of difference graphs with complete bipartite graphs, which has applications in the recent notion of local dimension of p…
In this note we show every orientation of a connected cubic graph admits an oriented 8-colouring. This lowers the best-known upper bound for the chromatic number of the family of orientations of connected cubic graphs. We further show that every such oriented graph admits a 2-dipath 7-colouring. These results imply that either the chromatic number for the family of oriented connected cubic graphs equals the 2-dipath chromatic number or the long-standing conjecture of Sopena [Journal of Graph Theory 25:191-205 1997] regarding the chromatic number of orientations of connected cubic graphs is fal…
Given two $3$-graphs $F$ and $H$, an $F$-covering of $H$ is a collection of copies of $F$ in $H$ such that each vertex of $H$ is contained in at least one copy of them. Let {$c_2(n,F)$} be the maximum integer $t$ such that every 3-graph with minimum codegree greater than $t$ has an $F$-covering. In this note, we answer an open problem of Falgas-Ravry and Zhao (SIAM J. Discrete Math., 2016) by determining the exact value of {$c_2(n, K_4^-)$} and {$c_2(n, K_5^-)$}, where $K_t^-$ is the complete $3$-graph on $t$ vertices with one edge removed.
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…
In these notes we investigate BIBDs with $λ=1$ that present subdesigns evenly covering both blocks and vertices: we determine some of their basic properties, consequence of already existing results in the literature, with regards to their size and the number of intersections of pairs and triples of subdesigns of a specific kind. We also describe the link between these particular BIBDs and the graph isomorphism problem, based on Babai's paper, and point out the characteristics of these designs that would lead to improvements of the algorithm for the GIP.
Let $G$ be a simple graph and let $p$ and $q$ be two integer-valued functions on $V(G)$ with $p< q$ in which for each $v\in V(G)$, $q(v) \ge \frac{1}{2}d_G(v)$ and $p(v) \ge \frac{1}{2} q(v)-2$. In this note, we show that $G$ has an orientation such that for each vertex $v$, $d^+_G(v)\in\{p(v),p(v)+1,q(v)-1,q(v)\}$ if and only if it has an orientation such that for each vertex $v$, $p(v) \le d^+_G(v)\le q(v)$ where $d^+_G(v)$ denotes the out-degree of $v$ in $G$. From this result, we refine a result due to Addario-Berry, Dalal, and Reed (2008) in bipartite simple graphs on the existence of deg…
In this note we show that any embedded graph has a checkerboard colourable twisted dual and any Eulerian embedded graph has a checkerboard colourable partial Petrial, answering questions posed by Ellis-Monaghan and Moffatt. The proofs are based on orientations of their medial graphs and orientations of boundary components of their corresponding ribbon graphs. The arrow presentations of ribbon graphs are also used. We also obtain two related results.
Given two $k$-uniform hypergraphs $F$ and $G$, we say that $G$ has an $F$-covering if every vertex in $G$ is contained in a copy of $F$. For $1\le i \le k-1$, let $c_i(n,F)$ be the least integer such that every $n$-vertex $k$-uniform hypergraph $G$ with $δ_i(G)> c_i(n,F)$ has an $F$-covering. The covering problem has been systematically studied by Falgas-Ravry and Zhao [Codegree thresholds for covering 3-uniform hypergraphs, SIAM J. Discrete Math., 2016]. Last year, Falgas-Ravry, Markström, and Zhao [Triangle-degrees in graphs and tetrahedron coverings in 3-graphs, Combinatorics, Probability a…
The concept of the brush number $b_r(G)$ was introduced for a simple connected undirected graph $G$. The concept will be applied to the Mycielskian graph $μ(G)$ of a simple connected graph $G$ to find $b_r(μ(G))$ in terms of an \emph{optimal orientation} of $G$. We prove a surprisingly simple general result for simple connected graphs on $n \geq 2$ vertices namely: $b_r(μ(G))= b_r(μ^{\rightarrow}(G)) = 2\sum\limits_{i=1}^{n}d^+_{G^{\rightarrow}_{b_r(G)}}(v_i).$
In this note we study the complexity of some generalizations of the notion of $st$-numbering. Suppose that given some functions $f$ and $g$, we want to order the vertices of a graph such that every vertex $v$ is preceded by at least $f(v)$ of its neighbors and succeeded by at least $g(v)$ of its neighbors. We prove that this problem is solvable in polynomial time if $fg\equiv 0$, but it becomes NP-complete for $f\equiv g \equiv 2$. This answers a question of the first author posed in 2009.
