Note on (semi-)proper orientation of some triangulated planar graphs

doi:10.1016/j.amc.2020.125723

Applied Mathematics and Computation

Volume 392, 1 March 2021, 125723

https://doi.org/10.1016/j.amc.2020.125723 Get rights and content

Highlights

•
We discuss the semi-proper orientation number and the proper orientation number of some triangulated planar graphs.
•
Semi-proper orientation number, which was first introduced by Dehghan and Havet in 2020, is based on proper orientation number.
•
Proper orientation number was first introduced by Araújo, Cohen, Rezende, Havet and Moura in 2015.
•
We attack the problem of proper orientation number proposed by Araújo, Cohen, Rezende, Havet and Moura in 2015 by proving tight upper bounds of the proper orientation number on triangulated grid and fish.
•
We prove the tight lower bound of the semi-proper orientation number on iterated triangulation.

Abstract

A weighted orientation of a graph G is a function (D, w) with an orientation D of G and with a weight function $w : E (G) \to Z_{+}$ . The in-weight $w_{D}^{-} (v)$ of a vertex v in D is the value $Σ_{u \in N_{D}^{-} (v)} w (u v)$ . A weighted orientation (D, w) of G is a semi-proper orientation if $w_{D}^{-} (v) \neq w_{D}^{-} (u)$ for all uv ∈ E(G). The semi-proper orientation number of G is defined as ${\vec{χ}}_{s} (G) = \min_{(D, w) \in Γ} \max_{v \in V (G)} w_{D}^{-} (v),$ where $Γ$ is the set of semi-proper orientations of G. When $w (e) = 1$ for any e ∈ E(G), this parameter is equal to the proper orientation number of G.

Dehghan and Havet (2007) introduced this parameter. Inspired by Araújo et al. (2019), we want to generalize some problems in Araújo et al. (2015) about proper orientation to the semi-proper version. In this paper, we study the (semi-)proper orientation number of some triangulated planar graphs.

Introduction

In a digraph, the notation (u, v) means an arc with tail u and head v. An orientation D of a graph G is a digraph obtained from G by replacing each edge uv of G by excactly one of the arcs (u, v) and (v, u). A weighted orientation of a graph G is a function (D, w) with an orientation D of G and with a weight function $w : E (G) \to Z_{+}$ . The in-weight $w_{D}^{-} (v)$ of a vertex v in D is the value $Σ_{u \in N_{D}^{-} (v)} w (u v)$ and we denote $w_{D}^{-} (U) = \sum_{v \in U} w_{D}^{-} (v)$ for any U⊆V. Whenever it is clear from the context, the subscript D will be omitted. A weighted orientation (D, w) of G is a semi-proper orientation if $w_{D}^{-} (v) \neq w_{D}^{-} (u)$ for all uv ∈ E(G). The semi-proper orientation number of G is defined as ${\vec{χ}}_{s} (G) = \min_{(D, w) \in Γ} \max_{v \in V (G)} w_{D}^{-} (v),$ where $Γ$ is the set of semi-proper orientations of G. An optimal semi-proper orientation is a semi-proper orientation (D, w) such that $\max_{v \in V (G)} w_{D}^{-} (v) = {\vec{χ}}_{s} (G)$ . If $w (e) = 1$ for any e ∈ E(G), then the in-weight of v is just the in-degree of v, and the orientation is just a proper orientation. We use $d_{D}^{-} (v)$ to denote the in-degree of v and use $\vec{χ} (G)$ to denote the proper orientation number. Likewise, an optimal proper orientation is a proper orientation D such that $\max_{v \in V (G)} d_{D}^{-} (v) = \vec{χ} (G)$ .

Araújo et al. [1] asked for a tight upper bound of the proper orientation number of planar graphs. Then in [3], Araújo et al. pointed out that the proper orientation number has non-monotonicity, i.e. for some (induced) subgraph H of G we have $\vec{χ} (H) > \vec{χ} (G)$ . This implies that even for some small classes of graphs, the upper bounds are not easy to get.

Henceforth, Araújo et al. [4] introduced the weighted proper orientation number. They asked to generalize some known results from the proper one to the weighted one, and also some open problems in [1] about proper orientation to the weighted version. Inspired by this, we want to generalize some problems in [1] about proper orientation to the semi-proper version, which may give us some new methods to solve problems in the proper version.

It is clear that ${\vec{χ}}_{s} (G)$ is not bigger than $\vec{χ} (G)$ for the same graph G. In [7], Dehgham and Havet showed that ${\vec{χ}}_{s} (G)$ is monotonic, which makes this parameter easier to study than the proper orientation number. Then they proved that ${\vec{χ}}_{s} (G) \leq 6$ for any planar graph G and ${\vec{χ}}_{s} (G) \geq ⌈ \frac{Mad (G)}{2} ⌉$ for any graph G, where $Mad (G) = \max_{H \subseteq G} \frac{2 | E (H) |}{| V (H) |}$ .

For finding a better lower bound of ${\vec{χ}}_{s} (G)$ for planar graphs we study the semi-proper orientation number of iterated triangulations. Then together with [8], we get a new method that is used to prove the tightness of the upper bounds of the proper orientation number of triangulated grids and fishes. The main result of this paper is the following.

Theorem 1

(i)
For any iterated triangulation Tr(n) with n ≥ 3, we have ${\vec{χ}}_{s} (Tr (n)) \geq 5$ and the bound is tight.
(ii)
For any triangulated grid Gr(n), we have $\vec{χ} (Gr (n)) \leq 4$ and the bound is tight.
(iii)
For any fish F_x,y(k), we have $\vec{χ} (F_{x, y} (k)) \leq 4$ and the bound is tight.

We will give the definitions of these three triangulated planar graphs and the proof of Theorem 1 in the next two sections, respectively.

Section snippets

Preliminaries

Only simple graphs are considered in this paper. We follow [6] for undefined notation and terminology. When X is a subset of the vertices of a digraph D, we denote by D[X] the subdigraph induced by X. Let $[k] = {1, 2, \dots, k}$ . Then we will define some special triangulated planar graphs following [5] and provide some structural properties of them.

Iterated triangulations, denoted by Tr(n), are defined as follows. Set Tr(0) $= K_{3}$ . For any i ≥ 0, we label all bounded faces of Tr(i) by F_j, for integer j ≥ 1,

Proof of Theorem 1

(i) By the definition of iterated triangulations, we have Tr(0) ⊂ Tr(1) ⊂ ⋅⋅⋅ ⊂ Tr(n). Since semi-proper orientation number is monotonic, it suffices to prove ${\vec{χ}}_{s} (Tr (3)) \geq 5$ . Label the vertices of Tr(3) as shown in Fig. 3.

By way of contradiction, suppose ${\vec{χ}}_{s} (Tr (3)) \leq 4$ . Let (D, w) be an optimal semi-proper orientation of Tr(3) and let V_i be the set of vertices in D with in-weight i ∈ {0, 1, 2, 3, 4}. Denote $W_{v} = Σ_{v \in V (G)} w_{D}^{-} (v)$ and $W_{e} = Σ_{e \in E (G)} w (e)$ . It is clear that $W_{v} = W_{e}$ . By the definition of the

Acknowledgment

The authors would like to thank Gregory Gutin for interesting discussions and the anonymous referees for their reports which helped to improve the presentation of the paper. Hui Lei was partially supported by the National Natural Science Foundation of China (No.12001296) and the Fundamental Research Funds for the Central Universities, Nankai University (No.63201163).

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There are more references available in the full text version of this article.

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An orientation of a graph $G$ is 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 in–out–proper orientation number of a graph $G$ , denoted by $\overset{\leftrightarrow}{χ} (G)$ , 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 $D$ . Borowiecki et al. proved that the in–out–proper orientation number is well-defined for any graph $G$ (Borowiecki et al., 2012). So we have $\overset{\leftrightarrow}{χ} (G) \leq Δ (G)$ , where $Δ (G)$ is the maximum degree of vertices in $G$ . We conjecture that there exists a constant number $c$ such that for every planar graph $G$ , we have $\overset{\leftrightarrow}{χ} (G) \leq c$ . Toward this speculation, we show that for every tree $T$ we have $\overset{\leftrightarrow}{χ} (T) \leq 3$ and this bound is sharp. Next, we study the in–out–proper orientation number of subcubic graphs. By using the properties of totally unimodular matrices we show that there is a polynomial time algorithm to determine whether $\overset{\leftrightarrow}{χ} (G) \leq 2$ , for a given graph $G$ with maximum degree three. On the other hand, we show that it is NP-complete to decide whether $\overset{\leftrightarrow}{χ} (G) \leq 1$ for a given bipartite graph $G$ with maximum degree three. Finally, we study the in–out–proper orientation number of regular graphs.
Proper orientation, proper biorientation and semi-proper orientation numbers of graphs
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Note on (semi-)proper orientation of some triangulated planar graphs

Highlights

Abstract

Introduction

Section snippets

Preliminaries

Proof of Theorem 1

Acknowledgment

Theoret. Comput. Sci.

Theoret. Comput. Sci.

Theoret. Comput. Sci.

Proper orientation number of triangle-free bridgeless outerplanar graphs

J. Graph Th.