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Journal of Combinatorial Optimization

, Volume 36, Issue 1, pp 55–64 | Cite as

Minimum 2-distance coloring of planar graphs and channel assignment

  • Junlei Zhu
  • Yuehua Bu
Article

Abstract

A 2-distance k-coloring of a graph G is a proper k-coloring such that any two vertices at distance two get different colors. \(\chi _{2}(G)\)=min{k|G has a 2-distance k-coloring}. Wegner conjectured that for each planar graph G with maximum degree \(\Delta \), \(\chi _2(G) \le 7\) if \(\Delta \le 3\), \(\chi _2(G) \le \Delta +5\) if \(4\le \Delta \le 7\) and \(\chi _2(G) \le \lfloor \frac{3\Delta }{2}\rfloor +1\) if \(\Delta \ge 8\). In this paper, we prove that: (1) If G is a planar graph with maximum degree \(\Delta \le 5\), then \(\chi _{2}(G)\le 20\); (2) If G is a planar graph with maximum degree \(\Delta \ge 6\), then \(\chi _{2}(G)\le 5\Delta -7\).

Keywords

Planar graph 2-Distance coloring Maximum degree 

1 Introduction

2-distance coloring arises from a communications problem known as the Channel Assignment Problem. The Channel Assignment Problem is a problem of assigning channels to the transmitters in some optimal manner. In this problem, there are transmitters located in some geographic region. It is usual for some pairs of transmitters to interfere with each other. There can be various reasons for this. This situation can be modeled by a graph G whose vertices are the transmitters, i.e \(V(G)=\{v_1, v_2, \ldots , v_n\}\), and \(v_iv_j\in E(G)\) if \(v_i\) and \(v_j\) interfere with each other. The aim is then to assign frequencies or channels to the transmitters in a manner that permits clear reception of the transmitted signals. The Channel Assignment Problem with variations has been studied by the Federal Communications Commission, ATT Bell Labs, the National Telecommunications and Information Administration, and the Department of Defense. Interpreting channels as colors gives rise to graph coloring problems. The idea of studying channel assignment with the aid of graphs is due to Griggs and Yeh [6].

Throughout this paper, all graphs considered are simple and finite. For a planar graph G, we denote its vertex set, edge set, face set, maximum degree and minimum degree by V(G), E(G), F(G), \(\Delta (G)\) and \(\delta (G)\) (V, E, F, \(\Delta \) and \(\delta \) for short) respectively. For \(x\in V(G)\cup F(G)\), let \(d_{G}(x)\) (d(x) for short) denote the degree of x in G. A vertex of degree k (resp. at least k, at most k) will be called k-vertex (resp. \(k^{+}\)-vertex, \(k^{-}\)-vertex) A face of degree k (resp. at least k, at most k) will be called k-face (resp. \(k^{+}\)-face, \(k^{-}\)-face). Let t(v) and \(n_3(v)\) be the number of 3-faces incident with vertex v and 3-vertices adjacent to vertex v, respectively. A \([v_1v_2\ldots v_k]\)-face is a k-face with vertices \(v_1, v_2,\ldots v_k\) on its boundary. Undefined notations are referred to [2].

A 2-distance k-coloring of G is a function f: \(V(G)\rightarrow \{1,2,3\ldots ,k\}\) such that \(|f(x)-f(y)|\ge 1\) if x and y are at distance at most 2. \(\chi _{2}(G)\)=min{\(k \mid G\) has a 2-distance k-coloring} is called the 2-distance chromatic number of G. In 1977, Wegner [15] first investigated the 2-distance chromatic number of planar graphs. He proved that the 2-distance chromatic number of a planar graph with maximum degree \(\Delta =3\) can be at most 8 and conjectured that the upper bound could be reduced to 7, which has been confirmed by Thomassen [16]. Moreover, Montassier and Raspaud [13] proved that 7 colors are necessary by giving an example. For planar graphs with maximum degree \(\Delta \ge 4\), Wegner posed the following conjecture.

Conjecture 1.1

If G is a planar graph, then \(\chi _{2}(G)\le \Delta +5\) if \(4\le \Delta \le 7\) and \(\chi _{2}(G)\le \lfloor \frac{3\Delta }{2}\rfloor +1\) if \(\Delta \ge 8\).

Wegner also gave some examples to illustrate that these upper bounds can be attained. Until now, Conjecture 1.1 is still open. However, several upper bounds in terms of maximum degree \(\Delta \) have been proved as follows. In 1993, Jonas [9] proved that \(\chi _2(G)\le 8\Delta -22\) for planar graphs with \(\Delta \ge 7\). Using the ideas in [7, 9] proved that \(\chi _2(G)\le 9\Delta -19\) for planar graphs with \(\Delta \ge 5\). Agnarsson and Halldorsson [1] showed that for every planar graph G with maximum degree \(\Delta \ge 749\), \(\chi _{2}(G)\le \lfloor \frac{9}{5}\Delta \rfloor +2\). A similar result was obtained by Borodin et al. [3]. For every planar graph G with maximum degree \(\Delta \ge 47\), \(\chi _{2}(G)\le \lceil \frac{9}{5}\Delta \rceil +1\). Nearly at the same time, Heuvel and McGuinness [7] gave a general upper bound without maximum degree restriction. For every planar graph G with maximum degree \(\Delta \), \(\chi _{2}(G)\le 2\Delta +25\). The currently best known upper bound for planar graphs was obtained by Molloy and Salavatipour [12]. For every planar graph G with maximum degree \(\Delta \), \(\chi _{2}(G)\le \lceil \frac{5}{3}\Delta \rceil +78\). Also, Molloy and Salavatipour [12] showed that if G is a planar graph with maximum degree \(\Delta \ge 241\), then \(\chi _{2}(G)\le \lceil \frac{5}{3}\Delta \rceil +25\).

Recently, Zhu [4] proved that the 2-distance chromatic number of a planar graph with maximum degree \(\Delta =4\) can be at most 13.

In this paper, we investigate the 2-distance coloring of planar graph G with maximum degree \(\Delta \). Actually, we prove the following theorems.

Theorem 1.1

If G is a planar graphs with maximum degree \(\Delta \le 5\), then \(\chi _2(G)\le 20\).

Theorem 1.2

If G is a planar graphs with maximum degree \(\Delta \ge 6\), then \(\chi _2(G)\le 5\Delta -7\).

Theorem 1.1 improved the upper bound 26 given in [7]. Theorem  1.2 improved the upper bound \(2\Delta +25\) while \(6\le \Delta \le 10\) in [7]. We will show Theorems 1.1 and 1.2 by induction on the number of vertices and edges. For an edge uv, let G / uv be the graph obtained from G by contracting the edge uv. After the operation G / uv have been performed, the following proposition formulates the essential properties of the vertex degree.

Proposition 1.1

Let \(H=G/uv\) and \(v'\) be the vertex in H corresponding to the edge uv. Then we have
  1. (1)

    \(d_H(w)\le d_G(w)\) for each vertex \(w\in V(H)\setminus \{v'\}\) and \(d_H(v')=d_G(u)+d_G(v)-2-t_G(uv)\), where \(t_G(uv)\) is the number of 3-faces incident with the edge uv.

     
  2. (2)

    \(d_{H}(w,w')\le d_{G}(w,w')\) and \(d_{H}(w,v')\le d_{G}(w,u)\) for any vertices \(w, w'\in V(H)\setminus \{v'\}\).

     

2 Planar graphs with maximum degree at most 5

In this section, let G be a counterexample with the smallest number of vertices and edges. That is \(\chi _2(G)>20\). Then, for any planar graph H with \(\Delta (H)\le 5\) and \(|V(H)|+|E(H)|<|V(G)|+|E(G)|\), \(\chi _2(H)\le 20\). We first establish structural properties of G. Let \(C=\{1,2,\ldots ,20\}\).

Lemma 2.1

G is connected and \(\delta (G)\ge 3\).

Proof

Suppose G is disconnected, by the minimality of G, every component of G has a 2-distance 20-coloring and so is G. If G has a 1-vertex v and \(uv\in E(G)\), then by the minimality of G, \(G-v\) has a 2-distance 20-coloring \(\varphi \). Since v has at most \(\Delta \le 5\) colors cannot be used, then we can color v. If G has a 2-vertex v and \(N(v)=\{u,w\}\), then we contact the edge uv to a vertex \(v'\). Since the obtained graph G / uv is also a planar graph with maximum degree at most 5, by the minimality of G, G / uv has a 2-distance 20-coloring \(\varphi \). Color the vertex u by \(\varphi (v')\). The remaining vertices keep their colors. Since v has at most \(2\times 5=10\) colors cannot be used, we can color v by a color \(\alpha \in \varphi (v)\in C-\{\varphi (x)|x\in V(G), d_G(v,x)\le 2\}\), \(\varphi \) can be extended to a 2-distance 20-coloring of G, a contradiction. \(\square \)

Lemma 2.2

Every 3-vertex is adjacent to three 5-vertices.

Proof

Assume that 3-vertex v is adjacent to a \(4^{-}\)-vertex u. Contact the edge uv to a vertex \(v'\). By the minimality of G, G / uv has a 2-distance 20-coloring \(\varphi \). Color the vertex u by \(\varphi (v')\). The remaining vertices keep their colors. Since v has at most \(2\times 5+4=14\) colors cannot be used, \(\varphi \) can be extended to a 2-distance 20-coloring of G, a contradiction. \(\square \)

Lemma 2.3

G has neither 3-face incident with a 3-vertex nor 3-face incident with two 4-vertices.

Proof

Assume that 3-face [uvw] is incident with a 3-vertex v and \(v_1\) is another neighbor of v. Let \(G'=G-v+v_1w\). By the minimality of G, \(G'\) has a 2-distance 20-coloring \(\varphi \). Since the colors on vertices \(v_1,u\) and w are distinct, v has at most \(5+4+4=13\) colors cannot be used, \(\varphi \) can be extended to a 2-distance 20-coloring of G. Assume that 3-face [uvw] is incident with two 4-vertices u and w. By the minimality of G, \(G-uw\) has a 2-distance 20-coloring \(\varphi \). Erase the colors on vertices u and w. Since each of u and w has at most \(5+5+4+2=16\) colors cannot be used, \(\varphi \) can be extended to a 2-distance 20-coloring of G, a contradiction. \(\square \)

Lemma 2.4

Every 3-vertex is incident with at least two \(5^{+}\)-faces.

Proof

By Lemma 2.3, 3-vertex v is not incident with any 3-face. Assume that 3-vertex v is incident with two 4-faces \([vv_1uv_2]\) and \([vv_2wv_3]\). Let \(G'=G-v+v_1v_3\). By the minimality of G, \(G'\) has a 2-distance 20-coloring \(\varphi \). Since the colors on vertices \(v_1, v_2\) and \(v_3\) are distinct, v has at most \(4+5+4=13\) colors cannot be used, \(\varphi \) can be extended to a 2-distance 20-coloring of G, a contradiction. \(\square \)

Lemma 2.5

G has no 4-vertex incident with two adjacent 3-faces.

Proof

Assume that 4-vertex v is incident with two adjacent 3-faces \([v_1vv_2]\) and \([v_2vv_3]\), \(v_4\) is another neighbors of v. Let \(G'=G-v+v_2v_4\). By the minimality of G, \(\chi _2(G')\le 20\). Let \(\varphi \) be a 2-distance 20-coloring of \(G'\). Since the colors on vertices \(v_1, v_2, v_3\) and \(v_4\) are distinct, v has at most \(4+3+4+5=16\) colors cannot be used, \(\varphi \) can be extended to a 2-distance 20-coloring of G, a contradiction. \(\square \)

Lemma 2.6

Let v be a 5-vertex, then
  1. (1)

    \(t(v)\le 3\).

     
  2. (2)

    If \(t(v)=3\), then \(n_3(v)=0\).

     
  3. (3)

    If \(t(v)=2\), then \(n_3(v)\le 1\).

     
  4. (4)

    If \(t(v)=1\), then \(n_3(v)\le 2\).

     
  5. (5)

    \(n_3(v)\le 4\).

     
  6. (6)

    If v is incident with a 3-face \([vv_1v_2]\), where \(d(v_1)=4\), then \(t(v)=1\).

     

Proof

  1. (1)

    Assume that 5-vertex v is incident with four 3-faces \([v_1vv_2], [v_2vv_3], [v_3vv_4]\) and \([v_4vv_5]\). Let \(G'=G-v+v_1v_5\). By the minimality of G, \(G'\) has a 2-distance 20-coloring \(\varphi \). Since the colors on vertices \(v_1, v_2, v_3, v_4\) and \(v_5\) are distinct, v has at most \(4+3\times 3+4=17\) colors cannot be used, \(\varphi \) can be extended to a 2-distance 20-coloring of G, a contradiction.

     
  2. (2)

    Let v be a 5-vertex and \(t(v)=3\). If v is incident with three 3-faces \([v_1vv_2], [v_2vv_3]\) and \([v_3vv_4]\) which are adjacent to each other, then by Lemma 2.3, \(d(v_i)\ne 3\) for \(i=1,2,3,4\). Let \(v_5\) is another neighbor of v and \(d(v_5)=3\). Let \(G'=G-v+v_1v_5+v_4v_5\). Since \(d_{G'}(v_5)=4\le 5\), by the minimality of G, \(G'\) has a 2-distance 20-coloring \(\varphi \). Since the colors on vertices \(v_1, v_2, v_3, v_4\) and \(v_5\) are distinct, v has at most \(4+3+3+4+3=17\) colors cannot be used, \(\varphi \) can be extended to a 2-distance 20-coloring of G, a contradiction. Otherwise, by Lemma 2.3, \(n_3(v)=0\).

     
  3. (3)

    Let v be a 5-vertex and \(t(v)=2\). If the two 3-faces are not adjacent, then by Lemma 2.3, \(n_3(v)\le 1\). Assume that 5-vertex v is incident with two adjacent 3-faces \([v_1vv_2]\) and \([v_2vv_3]\). By Lemma 2.3, \(d(v_i)\ne 3\) for \(i=1,2,3\). Let \(v_4\) and \(v_5\) be another two neighbors of v and \(d(v_4)=d(v_5)=3\). Let \(G'=G-v+v_3v_4+v_4v_5+v_1v_5\). Since \(d_{G'}(v_4)=d_{G'}(v_5)=4\le 5\), by the minimality of G, \(G'\) has a 2-distance 21-coloring \(\varphi \). Since the colors on vertices \(v_1, v_2, v_3, v_4\) and \(v_5\) are distinct, v has at most \(4+3+4+3+3=17\) colors cannot be used, \(\varphi \) can be extended to a 2-distance 20-coloring of G, a contradiction.

     
  4. (4)

    Let 5-vertex v be incident to a 3-face \([v_1vv_2]\). By Lemma 2.3, \(d(v_i)\ne 3\) for \(i=1,2\). Let \(v_3, v_4\) and \(v_5\) be another three neighbors of v and \(d(v_3)=d(v_4)=d(v_5)=3\). Let \(G'=G-v+v_2v_3+v_3v_4+v_4v_5+v_1v_5\). Since \(d_{G'}(v_3)=d_{G'}(v_4)=d_{G'}(v_5)=4\le 5\), by the minimality of G, \(G'\) has a 2-distance 20-coloring \(\varphi \). Since the colors on vertices \(v_1, v_2, v_3, v_4\) and \(v_5\) are distinct, v has at most \(4+4+3+3+3=17\) colors cannot be used, \(\varphi \) can be extended to a 2-distance 20-coloring of G, a contradiction.

     
  5. (5)

    Assume that 5-vertex is adjacent to five 3-vertices \(v_1, v_2, v_3, v_4\) and \(v_5\). By Lemma 2.3, \(v_iv_{(i+1)mod 5} \notin E(G)\), \(1\le i\le 5\). Let \(G'=G-v+v_iv_{(i+1)mod5}\), \(1\le i\le 5\). Since \(d_{G'}(v_i)=4\le 5\) for \(1\le i\le 5\), by the minimality of G, \(G'\) has a 2-distance 20-coloring \(\varphi \). Since the colors on vertices \(v_1, v_2, v_3, v_4\) and \(v_5\) are distinct, v has at most \(3\times 5=15\) colors cannot be used, \(\varphi \) can be extended to a 2-distance 20-coloring of G, a contradiction.

     
  6. (6)

    Let \(v_1,v_2,v_3,v_4,v_5\) be v’s five neighbors in clockwise and \(v_1v_2\in E(G)\), where \(d(v_1)=4\). By Lemma 2.5, \(v_2v_3 \notin E(G)\) and \(v_1v_5 \notin E(G)\). Assume that \(v_3v_4 \in E(G)\). By the minimality of G, \(G-vv_1\) has a 2-distance 20-coloring \(\varphi \). Erase the colors on v and \(v_1\). Since v and \(v_1\) has at most 19 and 17 colors cannot be used respectively, \(\varphi \) can be extended to a 2-distance 20-coloring of G, a contradiction. Thus, other face contains uv is a \(5^{+}\)-face. Similarly, the other face contains vw is a \(5^{+}\)-face.Since the colors on vertices \(v_1, v_2, v_3, v_4\) and \(v_5\) are distinct, v has at most \(4+4+3+3+3=17\) colors cannot be used, \(\varphi \) can be extended to a 2-distance 20-coloring of G, a contradiction. Thus, \(v_3v_4 \notin E(G)\). Similarly, \(v_4v_5 \notin E(G)\). \(\square \)

     

Lemma 2.7

Every 5-face contains at most one 3-vertex.

Proof

By Lemma 2.2, G contains no adjacent 3-vertices. Thus, 5-face is incident with at most two 3-vertices. Assume that 5-face \([v_1v_2v_3v_4v_5]\) contains two 3-vertices \(v_1\) and \(v_3\). Contact vertices \(v_1\) and \(v_3\) to a new vertex v. Denote the the obtained graph by \(G'\). Note that \(d_{G'}(v)=5\), by the minimality of G, \(G'\) has a 2-distance 20-coloring \(\varphi \). Since the colors on vertices \(v_2, v, v_4\) and \(v_5\) are distinct, color vertex \(v_3\) by \(\varphi (v)\). Since \(v_1\) has at most \(3\times 5=15\) colors cannot be used, \(\varphi \) can be extended to a 2-distance 20-coloring of G, a contradiction. \(\square \)

Proof of Theorem 1.1

Since G is connected, we define a weight function w by \(w(x)=d(x)-4\) for \(x\in V\bigcup F\). By Euler’s formula \(|V|-|E|+|F|=2\) and formula \(\sum \limits _{v\in V} d(v)=2|E|=\sum \limits _{f\in F} d(f)\), we can derive \(\sum \limits _{x\in V\cup F} w(x)=-8.\) We then design appropriate discharging rules and redistribute weights accordingly. Once the discharging is finished, a new weight function \(w'\) is produced. During the process, the total sum of weights is kept fixed. It follows that \(\sum \limits _{x\in V\cup F} w'(x)=\sum \limits _{x\in V\cup F} w(x)=-8.\) However, we will show that after the discharging is complete, the new weight function \(w'(x)\ge 0\) for all \(x\in V\cup F\). This leads to the following obvious contradiction
$$\begin{aligned} 0\le \sum \limits _{x\in V\cup F} w'(x)=\sum \limits _{x\in V\cup F} w(x)=-8<0. \end{aligned}$$
\(\square \)
Discharging rules
R1

Every \(5^{+}\)-face gives \(\frac{1}{5}\) to each adjacent 3-face and \(\frac{1}{4}\) to each incident 3-vertex.

R2

Every 5-vertex gives \(\frac{1}{3}\) to each incident 3-face and \(\frac{1}{4}\) to each adjacent 3-vertex.

R3

Every 5-vertex v with \(t(v)=1\) gives additional \(\frac{1}{6}\) the 3-face.

Checking

\(w'(v)\ge 0, v\in V\). By Lemma 2.1, \(\delta (G)\ge 3\).

Case \(d(v)=3\)

By Lemma 2.2 and R2, v gets \(\frac{1}{4}\times 3=\frac{3}{4}\) from its neighbors. By Lemma 2.4 and R1, v gets at least \(\frac{1}{4}\times 2=\frac{1}{2}\) from its incident \(5^{+}\)-faces. Thus \(w'(v)\ge 3-4+\frac{3}{4}+\frac{1}{2}>0\).

Case \(d(v)=4\)

Since v does not give out or receive any charge of v and thus \(w'(v)=w(v)=0\).

Case \(d(v)=5\)

By Lemma 2.6(1), \(t(v)\le 3\). If \(t(v)=3\), then by Lemma 2.6(2), \(n_3(v)=0\). By R2, \(w'(v)=5-4-\frac{1}{3}\times 3=0\). If \(t(v)=2\), then by Lemma 2.6(3), \(n_3(v)\le 1\). By R2, \(w'(v)\ge 5-4-\frac{1}{3}\times 2-\frac{1}{4}>0\). If \(t(v)=1\), then by Lemma 2.6(4), \(n_3(v)\le 2\). By R2 and R3, \(w'(v)\ge 5-4-\frac{1}{3}-\frac{1}{4}\times 2-\frac{1}{6}=0\). If \(t(v)=0\), then by Lemma 2.6(5), \(n_3(v)\le 4\). By R2, \(w'(v)\ge 5-4-\frac{1}{4}\times 4=0\).

Checking

\(w'(f)\ge 0, f\in F\)

Case \(d(f)=3\)

By Lemma 2.3, f is incident with at least two 5-vertices. If f is incident to three 5-vertices, then by R2, f receives at least \(\frac{1}{3}\times 3=1\) from these 5-vertices. Thus, \(w(v')\ge 3-4+1=0\). Otherwise, by Lemma 2.6(6), each 5-vertex incident with f is incident with exactly one 3-face. By R2 and R3, f receives \(\frac{1}{3}+\frac{1}{6}=\frac{1}{2}\) from each incident 5-vertices. Thus, \(w(v')\ge 3-4+\frac{1}{2}\times 2=0\).

Case \(d(f)=4\)

Since no 4-face gives out or receives out any charge, \(w'(f)=w(f)=0\).

Case \(d(f)=5\)

By Lemma 2.7, f contains at most one 3-vertex. If f contains one 3-vertex, then by Lemma 2.3, f is adjacent to at most three 3-faces. Thus, \(w'(f)\ge 5-4-\frac{1}{4}-\frac{1}{5}\times 3>0\) by R1. If f contains no 3-vertex, then \(w'(f)\ge 5-4-\frac{1}{5}\times 5=0\) by R1.

Case \(d(f)\ge 6\)

If f contains no 3-vertex, then \(w'(f)\ge d(f)-4-\frac{1}{5}\times d(f)>0\) by R1. If f contains d(f) 3-vertex, then by Lemma 2.3, f is not adjacent to any 3-face. Thus, \(w'(f)\ge d(f)-4-\frac{1}{4}\times d(f)>0\) by R1. If f contains \(1\le t\le d(f)-1\) 3-vertices, then by Lemma 2.3, f is adjacent to at most \((d(f)-t-1)\) 3-faces. Thus, \(w'(f)\ge d(f)-4-\frac{1}{4}\times t-\frac{1}{5}\times (d(f)-t-1)>0\) by R1.

3 Planar graphs with maximum degree at least 6

In this section, let \(G''\) be a counterexample to The orem 1.2, i.e. \(\Delta (G'')\ge 6\) and \(\chi _2(G'')>5\Delta -7\). Let planar graph G satisfy that \(\Delta (G)\le \Delta (G'')=\Delta \), \(\chi _2(G)>5\Delta -7\) and let \(|V(G)|+|E(G)|\) be minimum. It’s easy to see that for every planar graph H, if \(|V(H)|+|E(H)|<|V(G)|+|E(G)|\), then \(\chi _2(H)\le 5\Delta -7\). Obviously, G is connected. Take \(C=\{1,2,\ldots ,5\Delta -7\}\).

Lemma 3.1

\(\delta (G)\ge 3\).

Proof

If G has a 1-vertex v, then by the minimality of G, \(\chi _2(G-v)\le 5\Delta -7\). Since v has at most \(\Delta \) colors cannot be used, then we can color v. If G has a 2-vertex v and \(N(v)=\{u,w\}\), then we contact the edge uv to a vertex \(v'\). By the minimality of G, \(\chi _2(G/uv)\le 5\Delta -7\). Let \(\varphi \) be a 2-distance \((5\Delta -7)\)-coloring of G / uv. Color the vertex u by \(\varphi (v')\). The remaining vertices keep their colors. Since v has at most \(2\Delta \) colors cannot be used, \(\varphi \) can be extended to a 2-distance \((5\Delta -7)\)-coloring of G, a contradiction. \(\square \)

Lemma 3.2

G has no adjacent \(4^{-}\)-vertices.

Proof

Assume that G has two adjacent \(4^{-}\)-vertices u and v. Let \(G'=G-uv\). By the minimality of G, \(\chi _2(G')\le 5\Delta -7\). Let \(\varphi \) be a 2-distance \((5\Delta -7)\)-coloring of \(G'\). Erase the colors on u and v. Since u and v has at most \((3\Delta +3)\) colors cannot be used respectively, \(\varphi \) can be extended to a 2-distance \((5\Delta -7)\)-coloring of G, a contradiction. \(\square \)

Lemma 3.3

G has no 3-vertex incident with a 3-face.

Proof

Assume that 3-vertex v is incident with a 3-face \([v_2vv_3]\) and \(v_1\) is another neighbor of v. Let \(G'=G-v+v_1v_3\). By the minimality of G, \(\chi _2(G')\le 5\Delta -7\). Let \(\varphi \) be a 2-distance \((5\Delta -7)\)-coloring of \(G'\). Since the colors on vertices \(v_1,v_2\) and \(v_3\) are distinct, v has at most \(\Delta +(\Delta -1)+(\Delta -1)=3\Delta -2\) colors cannot be used, \(\varphi \) can be extended to a 2-distance \((5\Delta -7)\)-coloring of G, a contradiction. \(\square \)

Lemma 3.4

G has no 3-vertex incident with two 4-faces.

Proof

Assume that 3-vertex v is incident with two 4-faces \([vv_1uv_2]\) and \([vv_2wv_3]\). Let \(G'=G-v+v_1v_3\). By the minimality of G, \(\chi _2(G')\le 5\Delta -7\). Let \(\varphi \) be a 2-distance \((5\Delta -7)\)-coloring of \(G'\). Since the colors on vertices \(v_1, v_2\) and \(v_3\) are distinct, v has at most \((\Delta -1)+\Delta +(\Delta -1)=3\Delta -2\) colors cannot be used, \(\varphi \) can be extended to a 2-distance \((5\Delta -7)\)-coloring of G, a contradiction. \(\square \)

Lemma 3.5

G has no 4-vertex incident with a 3-face that contains another \((\Delta -1)^{-}\)-vertex.

Proof

Assume that 4-vertex v is incident with a 3-face \([v_1vv_2]\) and \(d(v_1)\le \Delta -1\), \(v_3, v_4\) are another two neighbors of v. Let \(G'=G-v+v_1v_3+v_1v_4\). Note that \(d_{G'}(v_1)\le \Delta (G)\). By the minimality of G, \(\chi _2(G')\le 5\Delta -7\). Let \(\varphi \) be a 2-distance \((5\Delta -7)\)-coloring of \(G'\). Since the colors on vertices \(v_1, v_2, v_3\) and \(v_4\) are distinct, v has at most \((\Delta -2)+(\Delta -1)+\Delta +\Delta =4\Delta -3\) colors cannot be used, \(\varphi \) can be extended to a 2-distance \((5\Delta -7)\)-coloring of G, a contradiction. \(\square \)

Lemma 3.6

G has no 4-vertex incident with two adjacent 3-faces.

Proof

Assume that 4-vertex v is incident with two adjacent 3-faces \([v_1vv_2]\) and \([v_2vv_3]\), \(v_4\) is another neighbors of v. Let \(G'=G-v+v_2v_4\). By the minimality of G, \(\chi _2(G')\le 5\Delta -7\). Let \(\varphi \) be a 2-distance \((5\Delta -7)\)-coloring of \(G'\). Since the colors on vertices \(v_1, v_2, v_3\) and \(v_4\) are distinct, v has at most \((\Delta -1)+(\Delta -2)+(\Delta -1)+\Delta =4\Delta -4\) colors cannot be used, \(\varphi \) can be extended to a 2-distance \((5\Delta -7)\)-coloring of G, a contradiction. \(\square \)

Lemma 3.7

Every 5-vertex is incident with at most three 3-faces.

Proof

Assume that 5-vertex v is incident with four 3-faces \([v_1vv_2], [v_2vv_3], [v_3vv_4]\) and \([v_4vv_5]\). Let \(G'=G-v+v_1v_5\). By the minimality of G, \(\chi _2(G')\le 5\Delta -7\). Let \(\varphi \) be a 2-distance \((5\Delta -7)\)-coloring of \(G'\). Since the colors on vertices \(v_1, v_2, v_3, v_4\) and \(v_5\) are distinct, v has at most \(\Delta -1+3\times (\Delta -2)+\Delta -1=5\Delta -8\) colors cannot be used, \(\varphi \) can be extended to a 2-distance \((5\Delta -7)\)-coloring of G, a contradiction. \(\square \)

Proof of Theorem 1.2

Since G is connected, we define a weight function w by \(w(x)=d(x)-4\) for \(x\in V\bigcup F\). By Euler’s formula \(|V|-|E|+|F|=2\) and formula \(\sum \limits _{v\in V} d(v)=2|E|=\sum \limits _{f\in F} d(f)\), we can derive \(\sum \limits _{x\in V\cup F} w(x)=-8.\) We then design appropriate discharging rules and redistribute weights accordingly. Once the discharging is finished, a new weight function \(w'\) is produced. During the process, the total sum of weights is kept fixed. It follows that \(\sum \limits _{x\in V\cup F} w'(x)=\sum \limits _{x\in V\cup F} w(x)=-8.\) However, we will show that after the discharging is complete, the new weight function \(w'(x)\ge 0\) for all \(x\in V\cup F\). This leads to the following obvious contradiction
$$\begin{aligned} 0\le \sum \limits _{x\in V\cup F} w'(x)=\sum \limits _{x\in V\cup F} w(x)=-8<0. \end{aligned}$$

\(\square \)

Discharging rules
R1

Every \(5^{+}\)-face gives \(\frac{1}{2}\) to each incident 3-vertex.

R2

Every \(5^{+}\)-vertex gives \(\frac{1}{3}\) to each incident 3-face.

R3

Every \(\Delta \)-vertex incident with a 3-face f and a \(4^{+}\)-face \(f_1\) gives additional \(\frac{1}{6}\) to f, where f and \(f_1\) are two adjacent faces.

Checking

\(w'(v)\ge 0, v\in V\). By Lemma 3.1, \(\delta (G)\ge 3\).

Case \(d(v)=3\)

By Lemmas 3.3 and  3.4, v is incident with at least two \(5^{+}\)-faces. By R1, each of these faces gives \(\frac{1}{2}\) to v and \(w'(v)\ge 3-4+\frac{1}{2}\times 2=0\). Case \(d(v)=4\). Since v does not give out or receive any charge of v and thus \(w'(v)=w(v)=0\).

Case \(d(v)=5\)

By Lemma 3.7, \(t(v)\le 3\). By R2, v gives out at most \(3\times \frac{1}{3}=1\) to incident 3-faces and thus \(w'(v)\ge 5-4-1=0\).

Case \(d(v)=6\)

If \(\Delta \ge 7\), then by R2, v gives out at most \(6\times \frac{1}{3}=2\) to incident 3-faces and thus \(w'(v)\ge 6-4-2=0\). It remains to analyze \(\Delta =6\). Note that R3 can be applied at most \(2\times (6-t(v))\) times. (Denote R3 can be applied at most k times, then \(t(v)\le 4, k\le t(v)\), \(t(v)=5, k\le 2\), \(t(v)=6, k=0\).) By R2 and R3, v gives out at most \(\frac{1}{3}\times t(v)+\frac{1}{6}\times 2\times (6-t(v))=2\) and thus \(w'(v)\ge 6-4-2=0\).

Case \(d(v)=7\)

If \(\Delta \ge 8\), then by R2, v gives out at most \(7\times \frac{1}{3}=\frac{7}{3}\) to incident 3-faces and thus \(w'(v)\ge 7-4-\frac{7}{3}>0\). It remains to analyze \(\Delta =7\). Note that R3 can be applied at most \(2\times (7-t(v))\) times. (Denote R3 can be applied at most k times, then \(t(v)\le 4, k\le t(v)\), \(t(v)=5, k\le 4\), \(t(v)=6, k\le 2\), \(t(v)=7, k=0\).) By R2 and R3, v gives out at most \(\frac{1}{3}\times t(v)+\frac{1}{6}\times 2\times (7-t(v))=\frac{7}{3}\) and thus \(w'(v)\ge 7-4-\frac{7}{3}\ge \frac{2}{3}\).

Case \(d(v)\ge 8\)

By R2 and R3, v gives out at most \(\frac{1}{3}+\frac{1}{6}=\frac{1}{2}\) to each incident 3-face and thus \(w'(v)\ge d(v)-4-\frac{1}{2}\times d(v)\ge 0\).

Checking

\(w'(f)\ge 0, f\in F\).

Case \(d(f)=3\)

By Lemma 3.3, f is not incident with 3-vertex. By Lemma 3.5, v is incident with one 4-vertex and two \(\Delta \)-vertices or three \(5^{+}\)-vertices. If f is incident with one 4-vertex v and two \(\Delta \)-vertices \(v_1, v_2\), then by Lemma 3.6, the other face containing \(vv_1\) is a \(4^{+}\)-face. Thus, by R3, \(v_1\) gives additional \(\frac{1}{6}\) to f. So is \(v_2\). Thus, \(w(v')\ge 3-4+\frac{1}{3}\times 2+\frac{1}{6}\times 2=0\). If f is incident with three \(5^{+}\)-vertices, then by R2, \(w(v')\ge 3-4+\frac{1}{3}\times 3=0\).

Case \(d(f)=4\)

Since no 4-face gives out or receives any charge, \(w'(f)=w(f)=0\).

Case \(d(f)\ge 5\)

By Lemma 3.2, there is no adjacent 3-vertices and thus f is incident with at most \(\lfloor \frac{d(f)}{2}\rfloor \) 3-vertices. By R1, \(w'(f)\ge d(f)-4-\frac{1}{2}\lfloor \frac{d(f)}{2}\rfloor \ge 0\).

4 Conclusion

In this paper, we proved two new upper bounds for 2-distance coloring. Our proofs can be turned into a constructive method. Following is an example.

Let \(v_0\) be the central vertex of \(K_{1,5}\) and \(v_i\) be the end vertices of \(K_{1,5}\), \(i=1,2,\cdots ,5\).
  1. Step 1:

    Color v with color 1;

     
  2. Step 2:

    Color \(v_1\) with color 2, this is possible because \(2\ne 1\);

     
  3. Step 3:

    Color \(v_2\) with color 3, this is possible because \(3\notin \{1,2\}\);

     
  4. Step 4:

    Color \(v_3\) with color 4, this is possible because \(4\notin \{1,2,3\}\)

     
  5. Step 5:

    Color \(v_4\) with color 5, this is possible because \(5\notin \{1,2,3,4\}\);

     
  6. Step 6:

    Color \(v_5\) with color 6, this is possible because \(6\notin \{1,2,3,4,5\}\).

     
This constructive method can provide an approximation solution for the channel assignment problem with minimum frequency span.

Moreover, our approach can be used for study of other variations of coloring and labeling [5, 8, 10, 11, 14]

Notes

Acknowledgements

The research work was supported by NFSC 11771403.

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© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  1. 1.College of Mathematics, Physics and Information EngineeringJiaxing UniversityJiaxingChina
  2. 2.Department of MathematicsZhejiang Normal UniversityJinhuaChina
  3. 3.Zhejiang Normal University Xingzhi CollegeJinhuaChina

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