Minimum 2distance coloring of planar graphs and channel assignment
Abstract
A 2distance kcoloring of a graph G is a proper kcoloring such that any two vertices at distance two get different colors. \(\chi _{2}(G)\)=min{kG has a 2distance kcoloring}. 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 2Distance coloring Maximum degree1 Introduction
2distance 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 kvertex (resp. \(k^{+}\)vertex, \(k^{}\)vertex) A face of degree k (resp. at least k, at most k) will be called kface (resp. \(k^{+}\)face, \(k^{}\)face). Let t(v) and \(n_3(v)\) be the number of 3faces incident with vertex v and 3vertices adjacent to vertex v, respectively. A \([v_1v_2\ldots v_k]\)face is a kface with vertices \(v_1, v_2,\ldots v_k\) on its boundary. Undefined notations are referred to [2].
A 2distance kcoloring 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 2distance kcoloring} is called the 2distance chromatic number of G. In 1977, Wegner [15] first investigated the 2distance chromatic number of planar graphs. He proved that the 2distance 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 2distance chromatic number of a planar graph with maximum degree \(\Delta =4\) can be at most 13.
In this paper, we investigate the 2distance 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
 (1)
\(d_H(w)\le d_G(w)\) for each vertex \(w\in V(H)\setminus \{v'\}\) and \(d_H(v')=d_G(u)+d_G(v)2t_G(uv)\), where \(t_G(uv)\) is the number of 3faces incident with the edge uv.
 (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 2distance 20coloring and so is G. If G has a 1vertex v and \(uv\in E(G)\), then by the minimality of G, \(Gv\) has a 2distance 20coloring \(\varphi \). Since v has at most \(\Delta \le 5\) colors cannot be used, then we can color v. If G has a 2vertex 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 2distance 20coloring \(\varphi \). 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 2distance 20coloring of G, a contradiction. \(\square \)
Lemma 2.2
Every 3vertex is adjacent to three 5vertices.
Proof
Assume that 3vertex 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 2distance 20coloring \(\varphi \). 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 2distance 20coloring of G, a contradiction. \(\square \)
Lemma 2.3
G has neither 3face incident with a 3vertex nor 3face incident with two 4vertices.
Proof
Assume that 3face [uvw] is incident with a 3vertex v and \(v_1\) is another neighbor of v. Let \(G'=Gv+v_1w\). By the minimality of G, \(G'\) has a 2distance 20coloring \(\varphi \). 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 2distance 20coloring of G. Assume that 3face [uvw] is incident with two 4vertices u and w. By the minimality of G, \(Guw\) has a 2distance 20coloring \(\varphi \). 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 2distance 20coloring of G, a contradiction. \(\square \)
Lemma 2.4
Every 3vertex is incident with at least two \(5^{+}\)faces.
Proof
By Lemma 2.3, 3vertex v is not incident with any 3face. Assume that 3vertex v is incident with two 4faces \([vv_1uv_2]\) and \([vv_2wv_3]\). Let \(G'=Gv+v_1v_3\). By the minimality of G, \(G'\) has a 2distance 20coloring \(\varphi \). 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 2distance 20coloring of G, a contradiction. \(\square \)
Lemma 2.5
G has no 4vertex incident with two adjacent 3faces.
Proof
Assume that 4vertex v is incident with two adjacent 3faces \([v_1vv_2]\) and \([v_2vv_3]\), \(v_4\) is another neighbors of v. Let \(G'=Gv+v_2v_4\). By the minimality of G, \(\chi _2(G')\le 20\). Let \(\varphi \) be a 2distance 20coloring 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 2distance 20coloring of G, a contradiction. \(\square \)
Lemma 2.6
 (1)
\(t(v)\le 3\).
 (2)
If \(t(v)=3\), then \(n_3(v)=0\).
 (3)
If \(t(v)=2\), then \(n_3(v)\le 1\).
 (4)
If \(t(v)=1\), then \(n_3(v)\le 2\).
 (5)
\(n_3(v)\le 4\).
 (6)
If v is incident with a 3face \([vv_1v_2]\), where \(d(v_1)=4\), then \(t(v)=1\).
Proof
 (1)
Assume that 5vertex v is incident with four 3faces \([v_1vv_2], [v_2vv_3], [v_3vv_4]\) and \([v_4vv_5]\). Let \(G'=Gv+v_1v_5\). By the minimality of G, \(G'\) has a 2distance 20coloring \(\varphi \). 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 2distance 20coloring of G, a contradiction.
 (2)
Let v be a 5vertex and \(t(v)=3\). If v is incident with three 3faces \([v_1vv_2], [v_2vv_3]\) 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 \(G'=Gv+v_1v_5+v_4v_5\). Since \(d_{G'}(v_5)=4\le 5\), by the minimality of G, \(G'\) has a 2distance 20coloring \(\varphi \). 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 2distance 20coloring of G, a contradiction. Otherwise, by Lemma 2.3, \(n_3(v)=0\).
 (3)
Let v be a 5vertex and \(t(v)=2\). If the two 3faces are not adjacent, then by Lemma 2.3, \(n_3(v)\le 1\). Assume that 5vertex v is incident with two adjacent 3faces \([v_1vv_2]\) 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 \(G'=Gv+v_3v_4+v_4v_5+v_1v_5\). Since \(d_{G'}(v_4)=d_{G'}(v_5)=4\le 5\), by the minimality of G, \(G'\) has a 2distance 21coloring \(\varphi \). 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 2distance 20coloring of G, a contradiction.
 (4)
Let 5vertex v be incident to a 3face \([v_1vv_2]\). 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 \(G'=Gv+v_2v_3+v_3v_4+v_4v_5+v_1v_5\). Since \(d_{G'}(v_3)=d_{G'}(v_4)=d_{G'}(v_5)=4\le 5\), by the minimality of G, \(G'\) has a 2distance 20coloring \(\varphi \). 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 2distance 20coloring of G, a contradiction.
 (5)
Assume that 5vertex is adjacent to five 3vertices \(v_1, v_2, v_3, v_4\) and \(v_5\). By Lemma 2.3, \(v_iv_{(i+1)mod 5} \notin E(G)\), \(1\le i\le 5\). Let \(G'=Gv+v_iv_{(i+1)mod5}\), \(1\le i\le 5\). Since \(d_{G'}(v_i)=4\le 5\) for \(1\le i\le 5\), by the minimality of G, \(G'\) has a 2distance 20coloring \(\varphi \). 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 2distance 20coloring of G, a contradiction.
 (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, \(Gvv_1\) has a 2distance 20coloring \(\varphi \). 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 2distance 20coloring 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 2distance 20coloring of G, a contradiction. Thus, \(v_3v_4 \notin E(G)\). Similarly, \(v_4v_5 \notin E(G)\). \(\square \)
Lemma 2.7
Every 5face contains at most one 3vertex.
Proof
By Lemma 2.2, G contains no adjacent 3vertices. Thus, 5face is incident with at most two 3vertices. Assume that 5face \([v_1v_2v_3v_4v_5]\) contains two 3vertices \(v_1\) 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 2distance 20coloring \(\varphi \). 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 2distance 20coloring of G, a contradiction. \(\square \)
Proof of Theorem 1.1
 R1

Every \(5^{+}\)face gives \(\frac{1}{5}\) to each adjacent 3face and \(\frac{1}{4}\) to each incident 3vertex.
 R2

Every 5vertex gives \(\frac{1}{3}\) to each incident 3face and \(\frac{1}{4}\) to each adjacent 3vertex.
 R3

Every 5vertex v with \(t(v)=1\) gives additional \(\frac{1}{6}\) the 3face.
 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 \(w'(v)\ge 34+\frac{3}{4}+\frac{1}{2}>0\).
 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, \(w'(v)=54\frac{1}{3}\times 3=0\). If \(t(v)=2\), then by Lemma 2.6(3), \(n_3(v)\le 1\). By R2, \(w'(v)\ge 54\frac{1}{3}\times 2\frac{1}{4}>0\). If \(t(v)=1\), then by Lemma 2.6(4), \(n_3(v)\le 2\). By R2 and R3, \(w'(v)\ge 54\frac{1}{3}\frac{1}{4}\times 2\frac{1}{6}=0\). If \(t(v)=0\), then by Lemma 2.6(5), \(n_3(v)\le 4\). By R2, \(w'(v)\ge 54\frac{1}{4}\times 4=0\).
 Checking

\(w'(f)\ge 0, f\in F\)
 Case \(d(f)=3\)

By Lemma 2.3, f is incident with at least two 5vertices. If f is incident to three 5vertices, then by R2, f receives at least \(\frac{1}{3}\times 3=1\) from these 5vertices. Thus, \(w(v')\ge 34+1=0\). Otherwise, by Lemma 2.6(6), each 5vertex incident with f is incident with exactly one 3face. By R2 and R3, f receives \(\frac{1}{3}+\frac{1}{6}=\frac{1}{2}\) from each incident 5vertices. Thus, \(w(v')\ge 34+\frac{1}{2}\times 2=0\).
 Case \(d(f)=4\)

Since no 4face 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 3vertex. If f contains one 3vertex, then by Lemma 2.3, f is adjacent to at most three 3faces. Thus, \(w'(f)\ge 54\frac{1}{4}\frac{1}{5}\times 3>0\) by R1. If f contains no 3vertex, then \(w'(f)\ge 54\frac{1}{5}\times 5=0\) by R1.
 Case \(d(f)\ge 6\)

If f contains no 3vertex, then \(w'(f)\ge d(f)4\frac{1}{5}\times d(f)>0\) by R1. If f contains d(f) 3vertex, then by Lemma 2.3, f is not adjacent to any 3face. Thus, \(w'(f)\ge d(f)4\frac{1}{4}\times d(f)>0\) by R1. If f contains \(1\le t\le d(f)1\) 3vertices, then by Lemma 2.3, f is adjacent to at most \((d(f)t1)\) 3faces. Thus, \(w'(f)\ge d(f)4\frac{1}{4}\times t\frac{1}{5}\times (d(f)t1)>0\) 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 1vertex v, then by the minimality of G, \(\chi _2(Gv)\le 5\Delta 7\). Since v has at most \(\Delta \) colors cannot be used, then we can color v. If G has a 2vertex 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 2distance \((5\Delta 7)\)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 2distance \((5\Delta 7)\)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 \(G'=Guv\). By the minimality of G, \(\chi _2(G')\le 5\Delta 7\). Let \(\varphi \) be a 2distance \((5\Delta 7)\)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 2distance \((5\Delta 7)\)coloring of G, a contradiction. \(\square \)
Lemma 3.3
G has no 3vertex incident with a 3face.
Proof
Assume that 3vertex v is incident with a 3face \([v_2vv_3]\) and \(v_1\) is another neighbor of v. Let \(G'=Gv+v_1v_3\). By the minimality of G, \(\chi _2(G')\le 5\Delta 7\). Let \(\varphi \) be a 2distance \((5\Delta 7)\)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 2distance \((5\Delta 7)\)coloring of G, a contradiction. \(\square \)
Lemma 3.4
G has no 3vertex incident with two 4faces.
Proof
Assume that 3vertex v is incident with two 4faces \([vv_1uv_2]\) and \([vv_2wv_3]\). Let \(G'=Gv+v_1v_3\). By the minimality of G, \(\chi _2(G')\le 5\Delta 7\). Let \(\varphi \) be a 2distance \((5\Delta 7)\)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 2distance \((5\Delta 7)\)coloring of G, a contradiction. \(\square \)
Lemma 3.5
G has no 4vertex incident with a 3face that contains another \((\Delta 1)^{}\)vertex.
Proof
Assume that 4vertex v is incident with a 3face \([v_1vv_2]\) and \(d(v_1)\le \Delta 1\), \(v_3, v_4\) are another two neighbors of v. Let \(G'=Gv+v_1v_3+v_1v_4\). 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 2distance \((5\Delta 7)\)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 2distance \((5\Delta 7)\)coloring of G, a contradiction. \(\square \)
Lemma 3.6
G has no 4vertex incident with two adjacent 3faces.
Proof
Assume that 4vertex v is incident with two adjacent 3faces \([v_1vv_2]\) and \([v_2vv_3]\), \(v_4\) is another neighbors of v. Let \(G'=Gv+v_2v_4\). By the minimality of G, \(\chi _2(G')\le 5\Delta 7\). Let \(\varphi \) be a 2distance \((5\Delta 7)\)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 2distance \((5\Delta 7)\)coloring of G, a contradiction. \(\square \)
Lemma 3.7
Every 5vertex is incident with at most three 3faces.
Proof
Assume that 5vertex v is incident with four 3faces \([v_1vv_2], [v_2vv_3], [v_3vv_4]\) and \([v_4vv_5]\). Let \(G'=Gv+v_1v_5\). By the minimality of G, \(\chi _2(G')\le 5\Delta 7\). Let \(\varphi \) be a 2distance \((5\Delta 7)\)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 2distance \((5\Delta 7)\)coloring of G, a contradiction. \(\square \)
Proof of Theorem 1.2
\(\square \)
 R1

Every \(5^{+}\)face gives \(\frac{1}{2}\) to each incident 3vertex.
 R2

Every \(5^{+}\)vertex gives \(\frac{1}{3}\) to each incident 3face.
 R3

Every \(\Delta \)vertex incident with a 3face 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 \(w'(v)\ge 34+\frac{1}{2}\times 2=0\). 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 3faces and thus \(w'(v)\ge 541=0\).
 Case \(d(v)=6\)

If \(\Delta \ge 7\), then by R2, v gives out at most \(6\times \frac{1}{3}=2\) to incident 3faces and thus \(w'(v)\ge 642=0\). It remains to analyze \(\Delta =6\). Note that R3 can be applied at most \(2\times (6t(v))\) 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 \(\frac{1}{3}\times t(v)+\frac{1}{6}\times 2\times (6t(v))=2\) and thus \(w'(v)\ge 642=0\).
 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 3faces and thus \(w'(v)\ge 74\frac{7}{3}>0\). It remains to analyze \(\Delta =7\). Note that R3 can be applied at most \(2\times (7t(v))\) 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 \(\frac{1}{3}\times t(v)+\frac{1}{6}\times 2\times (7t(v))=\frac{7}{3}\) and thus \(w'(v)\ge 74\frac{7}{3}\ge \frac{2}{3}\).
 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 3face 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 3vertex. By Lemma 3.5, v is incident with one 4vertex and two \(\Delta \)vertices or three \(5^{+}\)vertices. If f is incident with one 4vertex 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, \(w(v')\ge 34+\frac{1}{3}\times 2+\frac{1}{6}\times 2=0\). If f is incident with three \(5^{+}\)vertices, then by R2, \(w(v')\ge 34+\frac{1}{3}\times 3=0\).
 Case \(d(f)=4\)

Since no 4face gives out or receives any charge, \(w'(f)=w(f)=0\).
 Case \(d(f)\ge 5\)

By Lemma 3.2, there is no adjacent 3vertices and thus f is incident with at most \(\lfloor \frac{d(f)}{2}\rfloor \) 3vertices. 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 2distance coloring. Our proofs can be turned into a constructive method. Following is an example.
 Step 1:
Color v with color 1;
 Step 2:
Color \(v_1\) with color 2, this is possible because \(2\ne 1\);
 Step 3:
Color \(v_2\) with color 3, this is possible because \(3\notin \{1,2\}\);
 Step 4:
Color \(v_3\) with color 4, this is possible because \(4\notin \{1,2,3\}\)
 Step 5:
Color \(v_4\) with color 5, this is possible because \(5\notin \{1,2,3,4\}\);
 Step 6:
Color \(v_5\) with color 6, this is possible because \(6\notin \{1,2,3,4,5\}\).
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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