On B-Chromatic Number of Sun Let Graph and Wheel Graph Families

Journal of the Egyptian Mathematical Society (2015) 23, 215–218

Egyptian Mathematical Society Journal of the Egyptian Mathematical Society

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SHORT COMMUNICATION On b-chromatic number of sun let graph and wheel graph families

J. Vernold Vivin a,*, M. Vekatachalam b a Department of Mathematics, University College of Engineering Nagercoil, Anna University, Tirunelveli Region, Nagercoil 629 004, Tamil Nadu, India b Department of Mathematics, RVS Educational Trust’s Group of Institutions, RVS Faculty of Engineering, Coimbatore 641 402, Tamil Nadu, India

Received 6 November 2013; revised 15 March 2014; accepted 27 May 2014 Available online 30 June 2014

KEYWORDS Abstract A proper coloring of the graph assigns colors to the vertices, edges, or both so that prox- b-coloring; imal elements are assigned distinct colors. Concepts and questions of graph coloring arise naturally Sun let graph; from practical problems and have found applications in many areas, including Information Theory Wheel graph; and most notably Theoretical Computer Science. A b-coloring of a graph G is a proper coloring of Middle graph; the vertices of G such that there exists a vertex in each color class joined to at least one vertex in Total graph and line graph each other color class. The b-chromatic number of a graph G, denoted by uðGÞ, is the maximal integer k such that G may have a b-coloring with k colors. In this paper, we obtain the b-chromatic

number for the sun let graph Sn, line graph of sun let graph LðSnÞ, middle graph of sun let graph MðSnÞ, total graph of sun let graph TðSnÞ, middle graph of wheel graph MðWnÞ and the total graph of wheel graph TðWnÞ.

1991 MATHEMATICS SUBJECT CLASSIFICATION: 05C15; 05C75; 05C76

ª 2014 Production and hosting by Elsevier B.V. on behalf of Egyptian Mathematical Society.

1. Introduction of V into independent sets of G. The minimum cardinality k for which G has a k-coloring is the chromatic number vðGÞ All graphs considered in this paper are nontrivial, simple and of G. The b-chromatic number uðGÞ [1–4] of a graph G is undirected. Let G be a graph with vertex set V and edge set E. the largest positive integer k such that G admits a proper k-coloring in which every color class has a representative adja- A k-coloring of a graph G is a partition P ¼fV1; V2; ...; Vkg cent to at least one vertex in each of the other color classes. * Corresponding author. Tel.: +91 9566326299. Such a coloring is called a b-coloring. The b-chromatic number E-mail addresses: [email protected], [email protected] was introduced by Irving and Manlove [1] by considering (J. Vernold Vivin), [email protected] (M. Vekatachalam). proper colorings that are minimal with respect to a partial Peer review under responsibility of Egyptian Mathematical Society. order deﬁned on the set of all partitions of VðGÞ. They have shown that determination of uðGÞ is NP-hard for general graphs, but polynomial for trees. There has been an increasing Production and hosting by Elsevier interest in the study of b-coloring since the publication of [1].

1110-256X ª 2014 Production and hosting by Elsevier B.V. on behalf of Egyptian Mathematical Society. http://dx.doi.org/10.1016/j.joems.2014.05.011 216 J. Vernold Vivin, M. Vekatachalam

Irving and Manlove [1] have also proved the following following way. The vertex set of TðGÞ is VðGÞ[EðGÞ. Two upper bound of uðGÞ vertices x; y of TðGÞ are adjacent in TðGÞ in case one of the following holds: (i) x; y are in VðGÞ and x is adjacent to y in G. (ii) uðGÞ 6 DðGÞþ1 ð1Þ x; y are in EðGÞ and x; y are adjacent in G. (iii) x is in VðGÞ; y is Kouider and Maheó [5] gave some lower and upper bounds in EðGÞ, and x; y are incident in G. for the b-chromatic number of the cartesian product of two graphs. Kratochvı´l et al. [6] characterized bipartite graphs 3. b-coloring on Sunlet Graph and its line, middle and total for which the lower bound on the b-chromatic number is graphs attained and proved the NP-completeness of the problem to decide whether there is a dominating proper b-coloring even for connected bipartite graphs with k ¼ DðGÞþ1. Theorem 1. Let n P 6. Then, the b-chromatic number of the sun Effantin and Kheddouci studied [7–9] the b-chromatic let graph is uðSnÞ¼4. number for the complete caterpillars, the powers of paths, cycles, and complete k-ary trees. Proof. Let Sn be the sun let graph on 2n vertices. Let Faik [10] was interested in the continuity of the b-coloring VðSnÞ¼fv1; v2; ...; vng[fu1; u2; ...; ung where vi’s are the ver- and proved that chordal graphs are b-continuous. tices of cycles taken in cyclic order and ui’s are pendant vertices Corteel et al. [11] proved that the b-chromatic number such that each viui is a pendant edge. Consider the following 4- problem is not approximable within 120=133 for any coloring ðc1; c2; c3; c4Þ of Sn, assign the color c1 to vn; c2 to v1; c3 >0, unless P ¼ NP. to v2; c4 to v3; c2 to v4; c4 to vn1; c3 to un; c4 to u1 and for Hoańg and Kouider characterized in [12], the bipartite 2 6 i 6 n 1, assign the color c1 to ui. For 5 6 i 6 n 2, if graphs and the P4-sparse graphs for which each induced sub- any, assign to vertex vi one of the allowed colors - such color graph H of G has uðHÞ¼vðHÞ. exists, because degðviÞ¼3. An easy check shows that this Kouider and Zaker [13] proposed some upper bounds for coloring is a b - coloring. Therefore, uðSnÞ P 4 (see Fig. 1). the b-chromatic number of several classes of graphs in function Since D S 3, using 1:1 , we get that u S 6 4. Hence, of other graph parameters (clique number, chromatic number, ðÞ¼n ð Þ ð nÞ u S 4, for all n P 6. h biclique number). ð nÞ¼ Kouider and El Sahili proved in [14] by showing that if G is a d-regular graph with girth 5 and without cycles of length 6, Theorem 2. Let n P 6. Then, the b-chromatic number of the line then uðGÞ¼d þ 1. graph of sun let graph is uðLðSnÞÞ ¼ 5. Jakovac and Klavzˇar [2], proved that the b-chromatic number of cubic graphs is four with the exception of Petersen Proof. Let VðSnÞ¼fv1; v2; ...; vng[fu1; u2; ...; ung and E S = e0 : 1 6 i 6 n e : 1 6 i 6 n 1 e where graph, K3;3, prism over K3, and sporadic with 10 vertices. ð nÞ f i g[fi g[fng 6 6 0 Effantin and Kheddouci [15] proposed a discussion on rela- ei is the edge viviþ1 ð1 i n 1Þ; en is the edge vnv1 and ei 6 6 tionships between this parameter and two other coloring is the edge viui ð1 i nÞ. By the definition of line graph 0 6 6 0 6 6 parameters (the Grundy and the partial Grundy numbers). VðLðSnÞÞ ¼ EðSnÞ¼fui : 1 i ng[fvi : 1 i n 1g[ 0 0 0 0 6 6 The property of the dominating nodes in a b-coloring is very fvng where vi and ui represents the edge ei and ei ð1 i nÞ interesting since they can communicate directly with each par- respectively. Consider the following 5-coloring ðc1; c2; c3; 0 0 0 tition of the graph. c4; c5Þ of LSðÞn , assign the color c5 to u1; c4 to u2; c5 to u3; c1 0 0 0 0 0 6 6 Recently, Vernold Vivin and Venkatachalam [4] have to u4; c2 to u5; c1 to u6; c3 to vn; c3 to v6 and for 1 i 5, assign 0 6 6 0 proved the b-chromatic number for corona of any two graphs. the color ci to vi. For 7 i n, assign the color c2 to ui. For The authors also investigated the b-chromatic number of star graph families [3].

2. Preliminaries

The nsun let graph on 2n vertices is obtained by attaching n pendant edges to the cycle Cn and is denoted by Sn. For any integer n P 4, the wheel graph Wn is the n-vertex graph obtained by joining a vertex v1 to each of the n 1 vertices fw1; w2; ...; wn1g of the cycle graph Cn1. The line graph [16] of a graph G, denoted by LðGÞ,isa graph whose vertices are the edges of G, and if u; v 2 EðGÞ then uv 2 EðLðGÞÞ if u and v share a vertex in G. Let G be a graph with vertex set VðGÞ and edge set EðGÞ. The middle graph of G, denoted by MðGÞ is deﬁned as follows. The vertex set of MðGÞ is VðGÞ[EðGÞ. Two vertices x; y of MðGÞ are adjacent in MðGÞ in case one of the following holds: (i) x; y are in EðGÞ and x; y are adjacent in G. (ii) x is in VðGÞ; y is in EðGÞ, and x; y are incident in G. Let G be a graph with vertex set VðGÞ and edge set EðGÞ. The total graph [16] of G, denoted by TðGÞ is deﬁned in the Figure 1 Sunlet Graph Sn. On b-chromatic number of sun let graph and wheel graph families 217

Figure 2 Line graph of Sunlet Graph LSðÞn . Figure 3 Middle graph of Sunlet Graph MSðÞn .

6 6 0 7 i n 1, if any, assign to vertex vi one of the allowed col- 0 ors - such color exists, because degðviÞ¼4. An easy check 4. b-coloring of the middle and total graph of a wheel graph shows that this coloring is a b - coloring. Therefore, uðLðSnÞÞ P 5. Theorem 5. Let n P 7. Then, the b-chromatic number of the Since DðÞ¼LðSnÞ 4, using ð1:1Þ, we get that uðLðSnÞ 6 5. middle graph of wheel graph is uðMðWnÞÞ ¼ n; n is number of Hence, uðLðSnÞÞ ¼ 5, for all n P 6 (see Fig. 2). h vertices in Wn.

Theorem 3. Let n P 9. Then, the b-chromatic number of the Proof. Let VWðÞ¼n fgv; v1; v2; ...vn1 and let VMWðÞ¼ðÞn middle graph of sun let graph is uðMðSnÞÞ ¼ 7. fgv; v1; v2; ...vn1 [fge1; e2; ...en1 [ fgu1; u2; ...un1 . Where ui is the vertex of MWðÞn corresponding to the edge viviþ1 of Proof. Let V S v ; v ; ...; v u ; u ; ...; u and ð nÞ¼f 1 2 ng[f 1 2 ng WnðÞ1 6 i 6 n 1 and ei is the vertex of MWðÞn corresponding E S e0 : 1 6 i 6 n e : 1 6 i 6 n 1 e where e ð nÞ¼f i g[f i g[f ng i to the edge vvi of WnðÞ1 6 i 6 n 1 . By the deﬁnition of mid- is the edge v v 1 6 i 6 n 1 ; e is the edge v v and e0 is the i iþ1ð Þ n n 1 i dle graph, the vertices v and fei : ð1 6 i 6 n 1Þg induces a cli- edge v u 1 6 i 6 n . By the deﬁnition of middle graph V i ið Þ que of order n in MðWnÞ. Therefore, uðMðWnÞÞ P n. Consider M S V S E S v : 1 6 i 6 n u : 1 6 i 6 n ð ð nÞÞ ¼ ð nÞ[ ð nÞ¼f i g[f i g[ the following n-coloring of MWðÞn . For 1 6 i 6 n 1, assign v0 : 1 6 i 6 n u0 : 1 6 i 6 n , where v0 and u0 represents f i g[f i g i i the color ci to ei and assign the color cn to v. For the edge e and e0 1 6 i 6 n respectively. Consider the follow- i i ð Þ 1 6 i 6 n 1, assign to vertex ui one of the allowed colors - ing 7-coloring c ; c ; c ; c ; c ; c ; c of MS , assign the color ð 1 2 3 4 5 6 7Þ ðÞn such color exists, because degðuiÞ¼6. For 1 6 i 6 n 1, if 0 0 0 0 0 c7 to vn; c6 to u1; c1 to v1; c3 to v1; c5 to u2; c2 to v2; c4 to v2; c3 to 0 0 0 0 v3; c7 to u3; c6 to v3; c1 to v4; c5 to u4; c4 to v4; c2 to v5; c7 to 0 0 0 0 0 0 u5; c6 to v5; c3 to u6; c1 to v6; c5 to v6; c4to u7; c2 to v7; c7 to v7; c3 0 0 6 6 to u8; c1 to v8; c6 to v8 and for 1 i n, assign to vertex ui one of the allowed colors - such color exists, because degðuiÞ¼1. For 9 6 i 6 n, if any, assign to vertex vi and to vertex ui one of the allowed colors - such color exists, because degðviÞ¼3. For 6 6 0 9 i n 1, assign to vertex vi one of allowed colors-such color exists, because degðuiÞ¼6. An easy check shows that this coloring is a b-coloring. Therefore, uðMðSnÞÞ P 7.

Since DðÞ¼MðSnÞ 6, using ð1:1Þ, we get that uðMðSnÞÞ 6 7. Hence, uðMðSnÞÞ ¼ 7, for all n P 9 (see Fig. 3). h

Theorem 4. Let n P 9. Then, the b-chromatic number of the total graph of sun let graph is uðTðSnÞÞ ¼ 7.

Proof. Consider the coloring of MSðÞn introduced on the proof of Theorem 3. An easy check shows that this coloring is a b-coloring of TSðÞn . Hence, uðTðSnÞÞ P 7, for all n P 9.

Since DðÞ¼TðSnÞ 6, using ð1:1Þ, we get that uðTðSnÞÞ 6 7. h Hence, uðTðSnÞÞ ¼ 7, for all n P 9 (see Fig. 4). Figure 4 Total graph of Sunlet Graph TSðÞn . 218 J. Vernold Vivin, M. Vekatachalam

Acknowledgment

The authors would like to express their sincere gratitude for the referee’s constructive comments which helped to improve the presentation of this paper.

References

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Let us assume that uðMðWnÞÞ is greater than n, there must some families of graphs, Discrete Math. 306 (2006) 617–623. [14] M. Kouider, A. El Sahili, About b-colouring of regular graphs, be at least n þ 1 vertices of degree n in MðWnÞ, all with distinct colors, and each adjacent to vertices of all of the other colors. Rapport de Recherche No 1432, CNRS-Universite´Paris Sud- LRI, 2006, 02/2006. For n P 6, this is a contradiction. Thus, we have 6 [15] B. Effantin, H. Kheddouci, Discussion on the (partial) Grundy uðMðWnÞÞ n. Hence, uðMðWnÞÞ ¼ n; 8 n P 7 (see and b-chromatic numbers of graphs, Utilitas Math. 80 (2009) h Fig. 5). 65–89. [16] F. Harary, Graph Theory, Narosa Publishing Home, New Theorem 6. Let n P 7. Then, the b-chromatic number of the Delhi, 1969. total graph of wheel graph is uðTðWnÞÞ ¼ n; n is number of vertices in Wn.

Proof. Consider the coloring of MWðÞn introduced on the proof of Theorem 5. An easy check shows that this coloring is a b-coloring of TWðÞn . Hence, uðTðWnÞÞ ¼ n; 8 n P 6 (see Fig. 6). h