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Randić index and coloring number of a graph
Discrete Applied Mathematics 178 (2014) 163–165
Contents lists available at ScienceDirect
Discrete Applied Mathematics journal homepage: www.elsevier.com/locate/dam
Note
Randić index and coloring number of a graph✩ Baoyindureng Wu ∗ , Juan Yan, Xiaojing Yang College of Mathematics and System Science, Xinjiang University, Urumqi, Xinjiang 830046, PR China
article
info
Article history: Received 5 December 2013 Received in revised form 20 June 2014 Accepted 25 June 2014 Available online 9 July 2014 Keywords: Randić index Chromatic number Coloring number Point-arboricity
abstract √
The Randić index of a graph G, denoted by R(G), is defined as the sum of 1/ d(u)d(v) for all edges uv of G, where d(u) denotes the degree of a vertex u in G. The coloring number col(G) of a graph G is the smallest number k for which there exists a linear ordering of the vertices of G such that each vertex is preceded by fewer than k of its neighbors. It is wellknown that χ (G) ≤ col(G) for any graph G, where χ (G) denotes the chromatic number of G. In this note, we show that for any graph G with at least one edge, col(G) ≤ 2R(G), with equality if and only if G is a complete graph, possibly with some additional isolated vertices. This extends a theorem of Hansen and Vukicević. © 2014 Elsevier B.V. All rights reserved.
1. Introduction The Randić index R(G) of a (molecular) graph G was introduced by Milan Randić [11] in 1975 as the sum of √d(u1)d(v) over √ all edges uv of G, where d(u) denotes the degree of a vertex u in G, i.e., R(G) = uv∈E (G) 1/ d(u)d(v). This index is quite useful in mathematical chemistry and has been extensively studied, see [7]. For some recent results on Randić index, we refer to [3,8–10]. The chromatic number of G, denoted by χ (G), is the smallest number of colors needed to color all vertices of G such that no pair of adjacent vertices gets the same color. Hansen and Vukicević [5] established the following relation between the Randić index and the chromatic number of a graph. Theorem 1.1 (Hansen and Vukicević [5]). Let G be a simple graph with chromatic number χ (G) and Randić index R(G). Then χ(G) ≤ 2R(G) and equality holds if G is a complete graph, possibly with some additional isolated vertices. As usual, δ(G) and ∆(G) denote the minimum degree and the maximum degree of G, respectively. The coloring number col(G) of a graph G is the least integer k such that G has a vertex ordering in which each vertex is preceded by fewer than k of its neighbors. The degeneracy of G, denoted by deg(G), is defined as deg(G) = max{δ(H ) : H ⊆ G}. It is well-known (see p. 8 in [6]) that for any graph G, col(G) = deg(G) + 1.
(1)
List coloring is an extension of coloring of graphs, introduced by Vizing [13] and independently, by Erdős et al. [4]. For each vertex v of a graph G, let L(v) denote a list of colors assigned to v . A list coloring is a coloring l of vertices of G such that l(v) ∈ L(v) and l(x) ̸= l(y) for any xy ∈ E (G), where v, x, y ∈ V (G). A graph G is k-choosable if for any list assignment L to each vertex v ∈ V (G) with |L(v)| ≥ k, there always exists a list coloring l of G. The list chromatic number χl (G) (or choice number) of G is the minimum k for which G is k-choosable. ✩ Research supported by NSFC (11161046) and by Xingjiang Talent Youth Project (2013721012).
∗
Corresponding author. Fax: +86 991 8580377. E-mail addresses: [email protected], [email protected] (B. Wu).
http://dx.doi.org/10.1016/j.dam.2014.06.024 0166-218X/© 2014 Elsevier B.V. All rights reserved.
164
B. Wu et al. / Discrete Applied Mathematics 178 (2014) 163–165
The well-known Brooks theorem on chromatic number can be extended as follows: for any graph G,
χ (G) ≤ χl (G) ≤ col(G) ≤ ∆(G) + 1.
(2)
The detail of the inequalities (2) can be found in a survey paper of Tuza [12] on list coloring. The aim of this note is to extend Theorem 1.1 as follows. Theorem 1.2. If G is a simple graph with at least one edge, then col(G) ≤ 2R(G), with equality if and only if G is a complete graph, possibly with some additional isolated vertices. From the relation (2), Theorem 1.1 and the following corollary can be easily deduced from Theorem 1.2. Corollary 1.3. If G is a simple graph with at least one edge, then
χl (G) ≤ 2R(G), with equality if and only if G is a complete graph, possibly with some additional isolated vertices. 2. The proof The proof of Theorem 1.2 is based on a theorem of Hansen and Vukicević [5]. Theorem 2.1 (Hansen and Vukicević [5]). Let G be a simple graph with Randić index R, minimum degree δ and maximum degree ∆. Let v be a vertex of G with degree equal to δ . Then 1
R(G) − R(G − υ) ≥
2
δ . △
As a consequence, if G has at least one edge and v is a vertex of smallest positive degree, then R(G) − R(G − v) > 0. Proof of Theorem 1.2. We may assume that G has no isolated vertices, and let n be the order of G. Since deg(G) = max{δ(H ) : H ⊆ G} ≥ δ(G), we consider two cases. Case 1. deg(G) = δ(G) In this case, deg(G) = δ(G) and col(G) = δ(G) + 1 by inequality (1). Since |E (G)| ≥
R(G) =
√
uv∈E (G)
1
d(u)d(v)
≥
1
uv∈E (G)
∆(G)
≥
∆(G) + (n − 1)δ(G)
1
2
∆(G)
,
∆(G)+(n−1)δ(G) 2
, (3)
and thus 2R(G) ≥
∆(G) + (n − 1)δ(G) n−1 = δ(G) + 1 ≥ δ(G) + 1 = col(G), ∆(G) ∆(G)
equality holds if and only if G is ∆-regular (by (3)) and has ∆ = n − 1 (by the last inequality of the above displayed formula), therefore G is the complete graph of order n. Case 2. deg(G) > δ(G) We prove by induction on n that col(G) < 2R(G) in this case. Assume for a contradiction that col(G) ≥ 2R(G), and suppose by induction that col(G − u) ≤ 2R(G − u) for every vertex u ∈ V (G). Let v ∈ V (G) with d(v) = δ(G). Because no subgraph G′ attaining δ(G′ ) = deg(G) can contain v , by Theorem 2.1 and by the choice of G, we have col(G) = col(G − v) ≤ 2R(G − v) < 2R(G). The proof of the theorem is completed.
For a real number x, ⌊x⌋ denotes the maximum integer not greater than x, while ⌈x⌉ denotes the minimum integer not less than x. Another well-studied graph invariant, called point-arboricity, was introduced by Chartand, Kronk and Wall [2]. The point-arboricity ρ(G) of a graph G is defined as the minimum number of subsets into which the vertex set of G can be partitioned so that each subset induces an acyclic subgraph. Hence for any graph G, ρ(G) ≤ χ (G) ≤ 2ρ(G). Chartrand and deg(G) Kronk [1] proved that for any graph G, ρ(G) ≤ 1 + ⌊ 2 ⌋. Since col(G) = deg(G) + 1,
1+
deg(G) 2
=
col(G) + 1 2
=
col(G) 2
.
Thus, the following result is immediate from Theorem 1.2. Corollary 2.2. For any simple graph G, ρ(G) ≤ ⌈R(G)⌉.
B. Wu et al. / Discrete Applied Mathematics 178 (2014) 163–165
165
One can check that ρ(Kn ) = ⌈ 2n ⌉ = R(G). It is an interesting problem to characterize all graphs G with ρ(G) = ⌈R(G)⌉. What can one say about a graph G with χ (G) = 2R(G) − 1 (or with col(G) = 2R(G) − 1)? Let n be an even positive integer and M be a perfect matching of Kn . One can see that if G = Kn − M, then col(G) = n − 1 = 2R(G) − 1. Acknowledgments The authors are grateful to the referees for helpful comments. Special thanks are due to one of the referees for suggesting a simplification for the proof of our main theorem on an earlier version of this note. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13]
G. Chartrand, H.V. Kronk, The point-arboricity of planar graphs, J. Lond. Math. Soc. 44 (1969) 612–616. G. Chartand, H.V. Kronk, C.E. Wall, The point-arboricity of a graph, Israel J. Math. 6 (1968) 169–175. T.R. Divnić, L.R. Pavlović, Proof of the first part of the conjecture of Aouchiche and Hansen about the Randić, Discrete Appl. Math. 161 (2013) 953–960. P. Erdős, A. Rubin, H. Taylor, Choosability in graphs, Congr. Numer. 26 (1979) 125–157. P. Hansen, D. Vukicević, On the Randic index and the chromatic number, Discrete Math. 309 (2009) 4228–4234. T.R. Jensen, B. Toft, Graph Coloring Prolems, A Wiley-Interscience Publication, John Wiley and Sons, Inc., New York, 1995. X. Li, I. Gutman, Mahtematical Aspects of Randić-Type Molecular Structure Descriptors, in: Mathematical Chemistry Monographs, No. 1, Kragujevac, 2006. X. Li, Y. Shi, On a relation between the Randić index and the chromatic number, Discrete Math. 310 (2010) 2448–2451. J. Liu, M. Liang, B. Cheng, B. Liu, A proof for a conjecture on the Randić index of graphs with diameter, Appl. Math. Lett. 24 (2011) 752–756. B. Liu, L.R. Pavlović, T.R. Divnić, J. Liu, M.M. Stojanović, On the conjecture of Aouchiche and Hansen about the Randić index, Discrete Math. 313 (2013) 225–235. M. Randić, On characterization of molecular branching, J. Am. Chem. Soc. 97 (1975) 6609–6615. Z. Tuza, Graph colorings with local constraints—a survey, Discuss. Math. Graph Theory 17 (1997) 161–228. V.G. Vizing, Coloring the vertices of a graph in prescribed colors, Diskret. Anal. 29 (1976) 3–10 (Russian).
Contents lists available at ScienceDirect
Discrete Applied Mathematics journal homepage: www.elsevier.com/locate/dam
Note
Randić index and coloring number of a graph✩ Baoyindureng Wu ∗ , Juan Yan, Xiaojing Yang College of Mathematics and System Science, Xinjiang University, Urumqi, Xinjiang 830046, PR China
article
info
Article history: Received 5 December 2013 Received in revised form 20 June 2014 Accepted 25 June 2014 Available online 9 July 2014 Keywords: Randić index Chromatic number Coloring number Point-arboricity
abstract √
The Randić index of a graph G, denoted by R(G), is defined as the sum of 1/ d(u)d(v) for all edges uv of G, where d(u) denotes the degree of a vertex u in G. The coloring number col(G) of a graph G is the smallest number k for which there exists a linear ordering of the vertices of G such that each vertex is preceded by fewer than k of its neighbors. It is wellknown that χ (G) ≤ col(G) for any graph G, where χ (G) denotes the chromatic number of G. In this note, we show that for any graph G with at least one edge, col(G) ≤ 2R(G), with equality if and only if G is a complete graph, possibly with some additional isolated vertices. This extends a theorem of Hansen and Vukicević. © 2014 Elsevier B.V. All rights reserved.
1. Introduction The Randić index R(G) of a (molecular) graph G was introduced by Milan Randić [11] in 1975 as the sum of √d(u1)d(v) over √ all edges uv of G, where d(u) denotes the degree of a vertex u in G, i.e., R(G) = uv∈E (G) 1/ d(u)d(v). This index is quite useful in mathematical chemistry and has been extensively studied, see [7]. For some recent results on Randić index, we refer to [3,8–10]. The chromatic number of G, denoted by χ (G), is the smallest number of colors needed to color all vertices of G such that no pair of adjacent vertices gets the same color. Hansen and Vukicević [5] established the following relation between the Randić index and the chromatic number of a graph. Theorem 1.1 (Hansen and Vukicević [5]). Let G be a simple graph with chromatic number χ (G) and Randić index R(G). Then χ(G) ≤ 2R(G) and equality holds if G is a complete graph, possibly with some additional isolated vertices. As usual, δ(G) and ∆(G) denote the minimum degree and the maximum degree of G, respectively. The coloring number col(G) of a graph G is the least integer k such that G has a vertex ordering in which each vertex is preceded by fewer than k of its neighbors. The degeneracy of G, denoted by deg(G), is defined as deg(G) = max{δ(H ) : H ⊆ G}. It is well-known (see p. 8 in [6]) that for any graph G, col(G) = deg(G) + 1.
(1)
List coloring is an extension of coloring of graphs, introduced by Vizing [13] and independently, by Erdős et al. [4]. For each vertex v of a graph G, let L(v) denote a list of colors assigned to v . A list coloring is a coloring l of vertices of G such that l(v) ∈ L(v) and l(x) ̸= l(y) for any xy ∈ E (G), where v, x, y ∈ V (G). A graph G is k-choosable if for any list assignment L to each vertex v ∈ V (G) with |L(v)| ≥ k, there always exists a list coloring l of G. The list chromatic number χl (G) (or choice number) of G is the minimum k for which G is k-choosable. ✩ Research supported by NSFC (11161046) and by Xingjiang Talent Youth Project (2013721012).
∗
Corresponding author. Fax: +86 991 8580377. E-mail addresses: [email protected], [email protected] (B. Wu).
http://dx.doi.org/10.1016/j.dam.2014.06.024 0166-218X/© 2014 Elsevier B.V. All rights reserved.
164
B. Wu et al. / Discrete Applied Mathematics 178 (2014) 163–165
The well-known Brooks theorem on chromatic number can be extended as follows: for any graph G,
χ (G) ≤ χl (G) ≤ col(G) ≤ ∆(G) + 1.
(2)
The detail of the inequalities (2) can be found in a survey paper of Tuza [12] on list coloring. The aim of this note is to extend Theorem 1.1 as follows. Theorem 1.2. If G is a simple graph with at least one edge, then col(G) ≤ 2R(G), with equality if and only if G is a complete graph, possibly with some additional isolated vertices. From the relation (2), Theorem 1.1 and the following corollary can be easily deduced from Theorem 1.2. Corollary 1.3. If G is a simple graph with at least one edge, then
χl (G) ≤ 2R(G), with equality if and only if G is a complete graph, possibly with some additional isolated vertices. 2. The proof The proof of Theorem 1.2 is based on a theorem of Hansen and Vukicević [5]. Theorem 2.1 (Hansen and Vukicević [5]). Let G be a simple graph with Randić index R, minimum degree δ and maximum degree ∆. Let v be a vertex of G with degree equal to δ . Then 1
R(G) − R(G − υ) ≥
2
δ . △
As a consequence, if G has at least one edge and v is a vertex of smallest positive degree, then R(G) − R(G − v) > 0. Proof of Theorem 1.2. We may assume that G has no isolated vertices, and let n be the order of G. Since deg(G) = max{δ(H ) : H ⊆ G} ≥ δ(G), we consider two cases. Case 1. deg(G) = δ(G) In this case, deg(G) = δ(G) and col(G) = δ(G) + 1 by inequality (1). Since |E (G)| ≥
R(G) =
√
uv∈E (G)
1
d(u)d(v)
≥
1
uv∈E (G)
∆(G)
≥
∆(G) + (n − 1)δ(G)
1
2
∆(G)
,
∆(G)+(n−1)δ(G) 2
, (3)
and thus 2R(G) ≥
∆(G) + (n − 1)δ(G) n−1 = δ(G) + 1 ≥ δ(G) + 1 = col(G), ∆(G) ∆(G)
equality holds if and only if G is ∆-regular (by (3)) and has ∆ = n − 1 (by the last inequality of the above displayed formula), therefore G is the complete graph of order n. Case 2. deg(G) > δ(G) We prove by induction on n that col(G) < 2R(G) in this case. Assume for a contradiction that col(G) ≥ 2R(G), and suppose by induction that col(G − u) ≤ 2R(G − u) for every vertex u ∈ V (G). Let v ∈ V (G) with d(v) = δ(G). Because no subgraph G′ attaining δ(G′ ) = deg(G) can contain v , by Theorem 2.1 and by the choice of G, we have col(G) = col(G − v) ≤ 2R(G − v) < 2R(G). The proof of the theorem is completed.
For a real number x, ⌊x⌋ denotes the maximum integer not greater than x, while ⌈x⌉ denotes the minimum integer not less than x. Another well-studied graph invariant, called point-arboricity, was introduced by Chartand, Kronk and Wall [2]. The point-arboricity ρ(G) of a graph G is defined as the minimum number of subsets into which the vertex set of G can be partitioned so that each subset induces an acyclic subgraph. Hence for any graph G, ρ(G) ≤ χ (G) ≤ 2ρ(G). Chartrand and deg(G) Kronk [1] proved that for any graph G, ρ(G) ≤ 1 + ⌊ 2 ⌋. Since col(G) = deg(G) + 1,
1+
deg(G) 2
=
col(G) + 1 2
=
col(G) 2
.
Thus, the following result is immediate from Theorem 1.2. Corollary 2.2. For any simple graph G, ρ(G) ≤ ⌈R(G)⌉.
B. Wu et al. / Discrete Applied Mathematics 178 (2014) 163–165
165
One can check that ρ(Kn ) = ⌈ 2n ⌉ = R(G). It is an interesting problem to characterize all graphs G with ρ(G) = ⌈R(G)⌉. What can one say about a graph G with χ (G) = 2R(G) − 1 (or with col(G) = 2R(G) − 1)? Let n be an even positive integer and M be a perfect matching of Kn . One can see that if G = Kn − M, then col(G) = n − 1 = 2R(G) − 1. Acknowledgments The authors are grateful to the referees for helpful comments. Special thanks are due to one of the referees for suggesting a simplification for the proof of our main theorem on an earlier version of this note. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13]
G. Chartrand, H.V. Kronk, The point-arboricity of planar graphs, J. Lond. Math. Soc. 44 (1969) 612–616. G. Chartand, H.V. Kronk, C.E. Wall, The point-arboricity of a graph, Israel J. Math. 6 (1968) 169–175. T.R. Divnić, L.R. Pavlović, Proof of the first part of the conjecture of Aouchiche and Hansen about the Randić, Discrete Appl. Math. 161 (2013) 953–960. P. Erdős, A. Rubin, H. Taylor, Choosability in graphs, Congr. Numer. 26 (1979) 125–157. P. Hansen, D. Vukicević, On the Randic index and the chromatic number, Discrete Math. 309 (2009) 4228–4234. T.R. Jensen, B. Toft, Graph Coloring Prolems, A Wiley-Interscience Publication, John Wiley and Sons, Inc., New York, 1995. X. Li, I. Gutman, Mahtematical Aspects of Randić-Type Molecular Structure Descriptors, in: Mathematical Chemistry Monographs, No. 1, Kragujevac, 2006. X. Li, Y. Shi, On a relation between the Randić index and the chromatic number, Discrete Math. 310 (2010) 2448–2451. J. Liu, M. Liang, B. Cheng, B. Liu, A proof for a conjecture on the Randić index of graphs with diameter, Appl. Math. Lett. 24 (2011) 752–756. B. Liu, L.R. Pavlović, T.R. Divnić, J. Liu, M.M. Stojanović, On the conjecture of Aouchiche and Hansen about the Randić index, Discrete Math. 313 (2013) 225–235. M. Randić, On characterization of molecular branching, J. Am. Chem. Soc. 97 (1975) 6609–6615. Z. Tuza, Graph colorings with local constraints—a survey, Discuss. Math. Graph Theory 17 (1997) 161–228. V.G. Vizing, Coloring the vertices of a graph in prescribed colors, Diskret. Anal. 29 (1976) 3–10 (Russian).