Generalised colouring sums of graphs

Transcription

1 PURE MATHEMATICS RESEARCH ARTICLE Generaised coouring sums of graphs Johan Kok 1, NK Sudev * and KP Chithra 3 Received: 19 October 015 Accepted: 05 January 016 First Pubished: 09 January 016 Corresponding author: NK Sudev, Department of Mathematics, Vidya Academy of Science & Technoogy, Thaakkottukara, Thrissur , India E-mai: sudevnk@gmaicom Reviewing editor: Gerad Wiiams, University of Essex, UK Additiona information is avaiabe at the end of the artice Abstract: The notion of the b-chromatic number of a graph attracted much research interests and recenty a new concept, namey the b-chromatic sum of a graph, denoted by φ (G), has aso been introduced Motivated by the studies on b-chromatic sum of graphs, in this paper we introduce certain new parameters such as χ-chromatic sum, χ + -chromatic sum, b + -chromatic sum, π-chromatic sum and π + -chromatic sum of graphs We aso discuss certain resuts on these parameters for a seection of standard graphs Subjects: Advanced Mathematics; Combinatorics; Discrete Mathematics; Mathematics Statistics; Science Keywords: chromatic number; χ-chromatic sum; χ + -chromatic sum; b-chromatic number; b-chromatic sum; b + -chromatic sum; Thue chromatic number; π-chromatic sum; π + - chromatic sum AMS subject cassifications: 05C6; 05C05; 05C0; 05C38 1 Introduction For genera notations and concepts in graph theory and digraph theory, we refer to Bondy and Murty (1976), Chartrand and Lesniak (000), Chartrand and Zhang (009), Gross and Yeen (006), Harary (1969), West (001) Uness mentioned otherwise, a graphs mentioned in this paper are non-trivia, simpe, connected, finite and undirected graphs Graph coouring has become a fertie research area since its introduction in the second haf of nineteenth century It has numerous theoretica and practica appications Let us first reca the fact that in a proper coouring of a graph G, no two adjacent vertices in G can have the same coour The ABOUT THE AUTHORS Johan Kok is registered with the South African Counci for Natura Scientific Professions (South Africa) as a professiona scientist in both Physica Science and Mathematica Science His main research areas are in Graph Theory and the reconstruction of motor vehice coisions NK Sudev has been working as a Professor (Associate) in the Department of Mathematics, Vidya Academy of Science and Technoogy, Thrissur, India, for the ast fifteen years His primary research areas are Graph Theory and Combinatorics KP Chithra is an independent researcher in Mathematics Her primary research area is aso Graph Theory PUBLIC INTEREST STATEMENT Graph coouring attracted wide interest among researchers since its introduction in the second haf of the nineteenth century A number of interesting extrema graph theoretic probems were researched we Coouring sums are more recent and aows for appications where coours may be technoogies of kind with some reation between the distinct technoogies It is envisaged that coour products and other mathematica reations between coours wi naturay foow as enhanced research fieds It is foreseen that the modeing of metaboic or artificia inteigent structures as coours within arger rea or virtua iving structures of which certain components are modeed as graphs wi revea interesting appications Coouring sums are extremey usefu in many practica probems in project management, communication, routing and transportation, assignments, distributions etc 016 The Author(s) This open access artice is distributed under a Creative Commons Attribution (CC-BY) 40 icense Page 1 of 11

2 minimum number of coours in a proper coouring of a graph G is caed the chromatic number of G, denoted by χ(g) Consider a proper k-coouring of a graph G and denote the set of k coours by ={c 1,, c k } Aso, consider the disjoint subsets of V(G), defined by V ci ={v j :v j c i, v j V(G), c i }, 1 i k k Ceary, we can see that V(G) = V ci The notion of the b-coouring of a graph and the parameter b-chromatic number, φ(g), of a graph G(V, E), has been introduced in Irving and Manove (1999) as foows Let G be a graph on n vertices, say v 1, v The b-chromatic number of G is defined as the maximum number k of coours that can be used to coour the vertices of G, such that we obtain a proper coouring and each coour i, with 1 i k, has at east one eement x i which is adjacent to a vertex of every coour j, 1 j i k Such a coouring is caed a b-coouring of G (see Effatin & Kheddouci, 003; Irving & Manove, 1999) The concept of b-chromatic number has attracted much attention and many studies have been made on this parameter (see Effatin & Kheddouci, 003; Irving & Manove, 1999; Kok & Sudev, in press; Kouider & Mahéo, 00; Vaidya & Isaac, 014, 015; Vivin & Vekatachaam, 015) Genera coouring sum of graphs The notion of the b-chromatic sum of a given graph G, denoted by φ (G), has been introduced in Lisna and Sunitha (015) as the minimum of sum of coours c(v) of v for a v V in a b-coouring of G using φ(g) coours Some resuts on b-chromatic sums proved in Lisna and Sunitha (015), which are reevant and usefu resuts in our present study, are isted beow Theorem 1 φ (P n )= 3 (Lisna & Sunitha, 015) The b-chromatic sum of a path P n, n is (n 1)+3, if n 5, n is odd, + 1, if n 6, n is even, 4, if n = 3,, if n {, 4} Theorem Lisna & Sunitha, 015 The b-chromatic sum of a cyce C n + 3, if n is even, n 4, φ 3 (C n )= (n 1)+3, if n is odd, 6, if n = 4 Theorem 3 Lisna & Sunitha, 015 The b-chromatic sum of a whee graph W n+1 is 3(n 1) + 7, if n is odd, φ (W n+1 )= + 7, if n is even, n 4, 9, if n = 4 Theorem 4 (Lisna & Sunitha, 015) For a compete bipartite graph K m,n assume without oss of generaity that m n, then φ (K m,n )=m + n This interesting new invariant motivates us for studying simiar concepts in graph coouring This eads us to define the concept of the genera coouring sum of graphs as foows Definition 5 Let ={c 1,, c k } aows a b-coouring of a given graph G Ceary, there are k! ways of aocating the coours to the vertices of G The coour weight of coour, denoted by θ(c i ), is the number of times a particuar coour c i is aocated to vertices Then, the coouring sum of a coouring of a given graph G, denoted by ω( ), is defined to be ω( ) = i θ(c i ) Page of 11

3 In view of { the above } definition, the b-chromatic sum of a graph G can be viewed as φ (G) =min i θ(c i ), where this sum varies over a b-coourings of G In view of Definition 5, in this paper we introduce certain other coouring sums of graphs simiar to the b-chromatic sum of graphs 3 χ-chromatic sum of certain graphs The notion of the χ-chromatic sum of a graph G with respect to a proper k-coouring of G is introduced as foows Definition 31 Let ={c 1,, c k } be a proper { coouring } of a graph G Then, the χ-chromatic sum of G, denoted by χ (G), is defined as χ (G) =min i θ(c i ) where the sum varies over a minimum proper coourings of G In the foowing discussion, we investigate the χ-chromatic sum of certain fundamenta graph casses First, we determine the χ-chromatic sum of path graphs in the foowing theorem Theorem 3 The χ-chromatic sum of a path P n 1, if n = 1, χ (P n )=, if n is even,, if n is odd 1 Proof Being a bipartite graph, the vertices of a path graph P n can be cooured using two coours, say c 1 and c Then, we need to consider the foowing cases (1) Assume that n = 1 Then, P n K 1 with a singe vertex say v 1 Coour this vertex by the coour c 1 Hence, θ(c 1 )=1 Therefore, χ (P n )=1 () Let n be an even integer Then, the vertices of path P n can be cooured aternativey by the coours c 1 and c and hence θ(c 1 )=θ(c )= n Therefore, χ (P n )=1 n + n = (3) Let n > 1 be an odd integer Without oss of generaity, abe the vertices of P n with odd subscripts by the coour c 1 and the vertices with even subscripts by the coour c Then, θ(c 1 )= n+1 and θ(c )= n 1 Therefore, χ (P n )=1 n+1 + n 1 = 1 In a simiar way, the χ-chromatic sum of a cyce graph C n can be determined as foows Theorem 33 The χ-chromatic sum of a cyce C n is χ (C n )=3 n Proof Let be a proper coouring of the cyce C n If n is even, must contain at east two coours, say c 1 and c and if n is odd, then must contain at east three coours, say c 1 and c 3 Then, we consider the foowing cases (1) Let n be an odd integer Now, we can assign the coour c 1 to the vertices having odd subscripts other than n, the coour c to the vertices having even subscripts and the coour c 3 to the vertex v n Hence θ(c 1 )=θ(c )= n 1 and θ(c )=1 Therefore, χ 3 (G) =1 n 1 + n = 3 n+1 () Let n be an even integer Then, as expained in the previous resut, we can assign the coour c 1 to the vertices having odd subscripts and the coour c to the vertices having even subscripts Hence θ(c 1 )=θ(c )= n Therefore, χ (C n )=1 n + n = 3 n Combining the above two cases, we have χ (C n )=3 n A whee graph, denoted by W n+1, is defined to be the join of a cyce C n and a trivia graph K 1 That is, W n+1 = C n + K 1 The χ-chromatic sum of a whee graph is determined in the foowing theorem Page 3 of 11

4 Theorem 34 The χ-chromatic sum of a whee graph W n+1 χ (W n+1 )= +11, if n is odd,, if n is even +6 Proof Let us denote the centra vertex of the whee W n+1 by v and the vertices of the outer cyce of W n+1 by v 1, v Let be a minima proper coouring of W n+1 Then, must contain three coours, say c 1, if n is even and it must contain four coours, say c 1, c 4, if n is odd Hence, we have the foowing two cases (1) Let n be an even integer Then, in the outer cyce, n vertices have coour c and the other n vertices have the coour c 1 But the centra vertex being adjacent to a vertices of the outer cyce must be cooured using a new coour say c 3 Therefore, θ(c 1 )=θ(c )= n and θ(c 3 )=1 Hence, χ (G) =1 n + n + 3 = +6 () Let n be an odd integer Then, in the outer cyce C n, n 1 vertices have coour c 1 and n 1 vertices have the coour c and the remaining one vertex has the coour c 3 As mentioned in the above case, the centra vertex v must be cooured using a new coour say c 4 Therefore, θ(c 1 )=θ(c )= n 1 and θ(c )=θ(c )=1 and hence χ 3 4 (G) =1 n 1 + n 1 The foowing resut describes the χ-chromatic sum of a compete graph K n = +11 Proposition 35 The χ-chromatic sum of a compete graph K n is χ (K n )= n(n+1) Proof We know that in a proper coouring of K n, every vertex has distinct coours That is, χ(k n )=n n Therefore, θ(c i )=1, for a 1 i n Hence, we have χ (K n )= i = n(n+1) The χ-chromatic sum of a compete bipartite graph is determined in the foowing resut Proposition 36 The χ-chromatic sum of a compete bipartite graph K m,n, m n is χ (K m,n )=m + n Proof Assume that G be the compete bipartite graph with a bipartition (X, Y) such that X Y As a bipartite graph, G is -coourabe Since X Y, abe every vertex in X by the coour c 1 and every vertex of Y by the coour c Hence, θ(c 1 )= X and θ(c )= Y Therefore, χ (G) = X + Y Let us now reca the definition of a Rasta graph defined in Kok, Sudev, and Sudev (in press) as foows Definition 37 (Kok et a, in press) For a -term sum set {t 1, t, t 3,, t } with t 1 > t > t 3 > > t > 1, define the directed graph G () with vertices V(G () )={v i,j :1 j t i,1 i } and the arcs, A(G () )={(v i,j, v (i+1),m ):1 i ( 1),1 j t i and 1 m t (i+1) } In Kok and Sudev (in press), it is shown that for a Rasta graph R corresponding to the underying graph of G () the chromatic number φ(r) = Assume, without oss of generaity, that t (i 1) t i i=i i=i if is even and t (i 1) t i if is odd Then, the χ-chromatic sum of R is determined in the foowing i=i i=i theorem Theorem 38 The χ-chromatic sum of a Rasta graph R t χ (i 1) + t i, if is even, (R) = t (i 1) + t i, if is odd Page 4 of 11

5 Proof (1) Let be an even integer Since a vertices corresponding to t (i 1), 1 i are non-adjacent and hence we can coour these vertices by c 1 Aso, the remaining vertices, corresponding to t i, 1 i are aso non-adjacent among themseves and these vertices can be cooured using the coour c That is, θ(c 1 )= t (i 1) and θ(c )= t i Therefore, in this case χ (G) = t (i 1) + t i () Let be an odd integer Then, as expained in the above case, the vertices corresponding to t (i 1) ;1 i are non-adjacent among themseves and hence we can coour these vertices by c 1 The remaining vertices corresponding to t ;1 i i are aso non-adjacent among themseves and hence we can coour these vertices by c Therefore, θ(c 1 )= and θ(c )= t i and hence χ (G) = t (i 1) + t i 4 The χ + -chromatic sum of certain graphs We now define a new coouring sum, namey χ + -chromatic sum of a given graph G as foows t (i 1) Definition 41 Let ={c 1,, c k } be a proper coouring { of a graph } G Then, the χ + -chromatic sum of a graph G, denoted by χ + (G), is defined as χ + (G) =max i θ(c i ), where the sum varies over a minimum proper coourings of G Anaogous to the studies on χ-chromatic sum of certain graphs, here we study the χ + -chromatic sum of the corresponding graphs Theorem 4 For n 1, the χ + -chromatic sum of a path P n 1, if n = 1, χ + (P n )=, if n is even,, if n is odd +1 Proof If n = 1, we can assign c 1 to its unique vertex, which shows that χ + (P n )=1 Hence, et n > 1 As stated earier, every path P n, n is -coourabe Then, we have to consider the foowing cases (1) If n is even, as mentioned in Theorem 3, the vertices can be cooured aternativey by the coours c 1 and c and hence in this case, χ + (P n )= () If n is odd, then the mutuay non-adjacent n 1 vertices are cooured by c 1 and the remaining mutuay non-adjacent vertices can be cooured by the coour c Therefore, χ + (P n )=1 n 1 + n+1 n+1 = +1 This competes the proof The foowing is an immediate consequence of Theorem 3 and Theorem 4 Coroary 43 For a path P n, n 1 it foows that, χ + (P n )=χ (P n ) if n = 1 or even, ese χ + (P n )=χ (P n )+1 In the foowing resut, et us determine the χ + -chromatic sum of cyces Theorem 44 The χ + -chromatic sum of a cyce C n χ + (C n )= { 5n 3, if n is even,, if n is odd Page 5 of 11

6 Proof As stated earier, if n is even, then C n is -coourabe and if n is odd, C n is 3-coourabe Then, we have to consider the foowing cases (1) Let n be an even integer Then, the vertices of C n can be aternativey cooured by two coours c 1 and c We can see that exacty n vertices in C have the coours c and c n 1 each Therefore, θ(c 1 )=θ(c )= n Therefore, χ + (C n )= () Let n be an odd integer Then, we can assign coour c 3 to n 1 vertices, coour c n 1 to vertices and coour c 1 to one vertex, which provides a 3-coouring such that θ(c 1 )=1, θ(c )=θ(c 3 )= n 1 Therefore, χ + (C n )=5 n = 5n 3 The foowing theorem describes the χ + -chromatic sum of a whee graph W n+1 Theorem 45 χ + (W n+1 )= The χ + -chromatic sum of a whee graph W n+1, if n is even,, if n is odd { 5n+ 7n 1 Proof Let v 1, v be the vertices of the outer cyce the whee graph and v be its centra vertex We have aready mentioned in Theorem 34 that if n is even, then W n+1 is 3-coourabe and if n is odd, then W n+1 is 4-coourabe Then, we have the foowing cases (1) Let n be an even integer Then, we can assign the coour c 3 to n vertices of the outer cyce, the coour c to the remaining n vertices of the outer cyce and the coour c 1 to the centra vertex Hence, θ(c 3 )=θ(c )= n and θ(c )=1 Therefore, χ + 1 (W n+1 )=3 n + n + 1 = 5n+ () Let n be an odd integer Then, we can assign coour c 3 to the n 1 coour c 3 to the n 1 non-adjacent vertices, coour c 3 c 4 to the centra vertex, so that we get θ(c 3 )=θ(c 4 )= n 1, θ(c )=1 and θ(c 1 have χ (W n+1 )=4 n n 1 non-adjacent vertices, assign for the remaining singe vertex and coour )=1 Therefore, we = 7n 1 The foowing resut is an obvious and straightforward resut on the χ + -chromatic sum of compete graphs Proposition 46 The χ + -chromatic sum of a compete graph K n χ + (K n )=χ (G) = n(n+1) Proof Note that χ(k n )=n and hence as mentioned in Theorem 38, a vertices have distinct coours n That is, we have θ(c i )=1; for a 1 i n Hence, χ + (K n )= i = n(n+1) An obvious and straightforward resut on the χ + -chromatic sum of compete bipartite graphs is given beow Theorem 47 Consider the χ + -chromatic sum of a compete bipartite graph K m,n, m n 1, χ + (K m,n )=m + n Proof Since n m the maximum sum is obtained by aocating coour c to the n non-adjacent vertices and c 1 to the m non-adjacent vertices So θ(c 1 )=n and θ(c )=m Therefore, χ + (K m,n )=m + n The χ + -chromatic sum of Rasta graph can be determined as in the foowing theorem Theorem 48 The χ + -chromatic sum of Rasta graph R χ + (R) = t (i 1) + t i, t (i 1) + t i, if is even, if is odd Page 6 of 11

7 Proof (1) Let be an even integer Since a n vertices, corresponding to t, for a 1 i (i 1) are non- adjacent, these vertices can be cooured using the coour c By the same reason, the coour c 1 is aocated to the vertices corresponding to t i, 1 i Hence, θ(c )= 1 t i and θ(c )= t (i 1) Hence, χ + (R) = t (i 1) + t i for the even vaues of n () If is an odd integer, then the n+1 mutuay non-adjacent vertices can be cooured using c and the remaining n 1 mutuay non-adjacent vertices can be cooured using c 1 Hence, χ + (R) = t (i 1) + t i, for the odd vaues of n 5 b + -Chromatic Sum of Certain Graphs Anaogous to the χ-chromatic sum and χ + -chromatic sum of graphs, we can aso define the b + chromatic sum as foows Definition 51 The b + -chromatic sum of a graph G, denoted by φ + (G), is defined as φ + (G) =max{ i θ(c i )}, where the sum varies over a minima b-coouring using φ(g) coours Now, for determining the respective vaues of φ + for different graph casses, we use the proof techniques foowed in Lisna and Sunitha (015) Reversing the coouring pattern expained in Lisna and Sunitha (015), we work out the b + -chromatic sum of given graph casses Hence, we have the foowing resuts Theorem 5 The b + -chromatic sum of a path P n, n φ + (P n )= 5n 3 5n, if n 5, n is odd,, if n 6, n is even, 5, if n = 3,, if n {, 4} Proof We know that a b-coouring of a path P n requires at most three coours If 1 < n 4, the b-chromatic number of P n is In this context, the foowing cases are to be considered (1) Let n be even That is, n =, 4 If n =, then, one of its two vertices has coour c 1 and the other vertex has coour c Hence, the b + -chromatic sum of P is = 3 If n = 4, Let 1 ={v } and } be the coour casses of the coours c 1 and c, respectivey, so that ={c 1 } is a b-coouring of P n Then, the b + -chromatic sum of P = 6 Combining these two cases, it foows that φ (P n )=φ + (P n )=, for n =, 4 () Let n = 3 Then, et 1 ={v } and }, so that ={c 1 } is a b + -coouring of P n Then, the b + -chromatic sum of P = 5 If n 5, the b-chromatic number of a path P n is 3 Hence, we have to consider the foowing cases (3) Let n 5 and n be odd Now, et ={c 1 } be a coouring on P n such that 1 ={v 3 } be the coour cass of the coour c 1, ={v, v 5, v 7 } be the coour cass of the coour c and 3, v 6 1 } be the coour cass of coour c 3 Ceary, this coouring is a b + -coouring of P n Then, we have θ(c 1 )=1, θ(c )= n 1 and θ(c 3 )= n 1 Hence, for n 5 and n is odd, φ + (P n )= 3 (n 1)+ 5n 3 (n 1)+1 = (4) Let n 5 and n be even Here, assume that ={c 1 } be a coouring on P n such that the coour casses 1, and 3 are exacty as defined in the previous case This coouring is obviousy a b + coouring of P n Then, it foows that θ(c 1 )=1, θ(c )= n n 6, k is even, φ + (P n )=3 n + n and θ(c 3 )= n Hence, for + 1 = 5n Page 7 of 11

8 Simiary, the b + -chromatic sum of a cyce C n is determined in the foowing theorem Theorem 53 The b + -chromatic sum of a cyce C n 6, if n = 4, φ + 5n 3 (C n )=, if n is odd,, if n is even, n 4 5n 6 Proof First, et n = 4 It is to be noted that the b-chromatic number of the cyce C 4 is, where the vertices v 1 and v 3 have coour c 1 and the vertices v and v 4 have the coour c Therefore, the b + -chromatic sum of C 4 is + 1 = 6 Next, assume that n 4 We know that the b-chromatic number of a cyce C n, n 4 is 3 Let ={c 1 } be a b-coouring of a given cyce C n Here, we have to consider the foowing cases (1) Let n be odd Now a b-coouring which forms the coour casses, 1 ={v 3 },, v 6 1 } and 3 ={v, v 5, v 7 }, yied the desirabe b-coouring such that θ(c 1 )=1, θ(c )= n 1 and θ(c 3 )= n 1 Therefore, here the b+ -chromatic sum 3 n 1 + n = 5n 3 () Let n be even Now, a b-coouring which forms the coour casses, 1 ={v 3, v n },, v 6 } and 3 ={v, v 5, v 7 1 }, yied the desirabe b-coouring such that θ(c 1 )=, θ(c )= n and θ(c n 3 )= Therefore, we have φ+ (C n )=3 n + n + = 5n 6 This competes the proof Now, the b + -chromatic sum of a whee graph W n+1 is determined in the foowing resut Theorem 54 The b + -chromatic sum of a whee graph W n+1 11, if n = 4, φ + 7n 1 (W n+1 )=, if n is odd,, if n is even, n 4 7n 4 Proof We have aready stated that the b-chromatic number of the cyce C 4 is 3 Therefore, a b-coouring of W 5 = C 4 + K 1 must contain 3 coours, say c 1 and c 3 Let the corresponding coour casses be 1 ={v}, } and 3 ={v }, where v is the centra vertex of the whee graph Then, θ(c 1 )=1, θ(c )= and θ(c 3 )= Hence, φ + (W 5 )= = 11 Next, assume that n 4 Then, every b-coouring of W n+1 must contain 4 coours Let ={c 1, c 4 } be the required coouring of G Then, we have to consider the foowing cases (1) Assume that n is odd Then, coour the vertices of W n+1 using the coours in in such a way that the corresponding coour casses are 1 ={v}, ={v 3 } 3, v 6 1 } and 4 ={v, v 5, v 7 } Therefore, we have θ(c 1 )=θ(c )=1 and θ(c 3 )= n 1 Then, we have φ + (W n+1 )=4 n n = 7n 1 and θ(c 4 )= n 1 () Assume that n is even Coour the vertices of W n+1 in such a way that the corresponding coour casses are 1 ={v}, ={v 3, v n } 3, v 6 } and 4 ={v, v 5, v 7 1 } Then, we have θ(c 1 )=1, θ(c )= and θ(c 3 )=θ(c 4 )= n Hence, φ + (W n+1 )=4 n + 3 n = 7n 4 The foowing theorem describes the φ + -chromatic number of a compete bipartite graph Theorem 55 The b + -chromatic sum of a compete bipartite graph K m,n, m n is φ + (K m,n )=m + n Proof The resut foows directy from the proof of Theorem 47 Page 8 of 11

9 The b-chromatic sum and the b + -chromatic sum of Rasta Graph R is determined in the theorem given beow Theorem 56 The b-chromatic sum of a Rasta graph R t φ (i 1) + t i, if is even, (R) = t (i 1) + t i, if is odd, and the b + -chromatic sum of R φ + (R) = t (i 1) + t i, t (i 1) + t i, if is even, if is odd Proof The proof foows directy from the proofs of Theorem 38 and 48 6 Two Thue chromatic sums of a path A finite sequence S =(q 1, q, q 3,, q t ) of symbos of any aphabet is known to be non-repetitive if for a its subsequences (r 1, r, r 3,, r m );1 m t, the condition r i r i, 1 i m, hods Let G be a simpe undirected graph on n vertices and et a minimum set of coours aow a proper vertex coouring of G If the sequence of vertex coours of any path of even and finite ength in G is non-repetitive, then this proper coouring is said to be a Thue coouring of G (see Aon, Grytczuk, Hauszczak & Riordan, 00) The Thue chromatic number of G, denoted π(g), is defined as the minimum number of coours required for a Thue coouring of G It is known that π(p 1 )=1, π(p )=π(p 3 )= and for n 4, π(p n )=3 Determining π (P n ) is a hard probem, hence the probem is very hard for graphs in genera The foowing emma is the motivation for our further discussions in this paper Lemma 61 Škrabu áková, in press Up to equivaence, there is exacty one non-repetitive 3-coouring of the cyce C 11 In view of this emma, we restrict our further discussion to the path P 11 Let the vertices of P n be abeed from eft to right to be v 1, v, v 11 Reca that the coouring sum of a coouring is defined by ω( ) = i θ(c i ) The possibe minimum Thue coourings of P 11 are isted beow (1) 1 ) and ω( 1 )= () ) and ω( )=1 (3) 3 ) and ω( 3 )=1 (4) 4 ) and ω( 4 )= (5) 5 ) and ω( 5 )=0 (6) 6 ) and ω( 6 )=1 Page 9 of 11

10 (7) 7 =(c ) and ω( 7 )=3 (8) 8 =(c ) and ω( 8 )=1 (9) 9 =(c ) and ω( 9 )=1 (10) 10 =(c ) and ω( 10 )=3 (11) 11 =(c ) and ω( 11 )= (1) 1 =(c ) and ω( 1 )=1 (13) 13 ) and ω( 13 )= (14) 14 ) and ω( 14 )=3 (15) 15 ) and ω( 15 )=3 (16) 16 ) and ω( 16 )= (17) 17 ) and ω( 17 )=4 (18) 18 ) and ω( 18 )=3 (19) 19 ) and ω( 19 )=1 (0) 0 ) and ω( 0 )= (1) 1 ) and ω( 1 )= () ) and ω( )=1 (3) 3 ) and ω( 3 )=0 (4) 4 ) and ω( 4 )=From the above ist, we note that π (P 11 )=0 and π + (P 11 )=4 We strongy beieve that the next conjecture hods Conjecture 6 For a path P n, n 4, there is a unique permutation over a proper b-coourings for which φ + (P n ) is obtained, and exacty two permutations for which φ (P n ) is obtained The foowing genera resut is of importance for a variations of coouring sums discussed thus far It hods for improper coourings as we A genera coouring which meets some genera coouring index is caed the θ-chromatic number of G and denoted, θ(g) Theorem 63 (Makungu s Theorem 1 ) If the set of coours ={c j :1 j k} aows a genera coouring, :f (v i )=c, {1,, 3,, k} of G, such that ω( ) =θ (G) =min{ i θ(c i ): -coourings of G}, then θ + (G) = i θ(c (k+1) i ) Proof Since for a 1 a it foows that 1 a 1 + a a a, it foows through immediate induction that if a 1 a a 3 a k then for permuted one-on-one aocations of the eements in b i {1,, 3,, k} to form a i b i we have, min{ a i b i }= i a i and max{ a i b i }= i a (k+1) i Hence, if a θ-coouring of G is aowed by ={c 1,, c k } such that, θ(c 1 ) θ(c ) θ(c 3 ) θ(c k ) then, θ (G) = i θ(c i ) and θ + (G) = i θ(c (k+1) i ) Page 10 of 11

11 Acknowedgements The authors gratefuy acknowedge the contributions of anonymous referees whose critica and constructive comments payed a vita roe in improving the quaity and presentation stye of the paper in a significant way Funding The authors received no direct funding for this research Author detais Johan Kok 1 E-mai: kokkiek@tshwanegovza ORCID ID: NK Sudev E-mai: sudevnk@gmaicom ORCID ID: KP Chithra 3 E-mai: chithrasudev@gmaicom ORCID ID: 1 Tshwane Metropoitan Poice Department, City of Tshwane, Repubic of South Africa Department of Mathematics, Vidya Academy of Science & Technoogy, Thaakkottukara, Thrissur, , India 3 Naduvath Mana, Nandikkara, Thrissur, , India Citation information Cite this artice as: Generaised coouring sums of graphs, Johan Kok, NK Sudev & KP Chithra, Cogent Mathematics (016), 3: Note 1 The first author dedicates this theorem to Makungu Mathebua and he hopes this young ady wi grow up with a deep fondness for mathematics References Aon, N, Grytczuk, J, Hauszczak, M, & Riordan, O (00) Non-repetitive coourings of graphs Random Structures & Agorithms, 1, doi:10100/rsa10057 Bondy, J A, & Murty, U S R (1976) Graph theory with appications London: Macmian Press Chartrand, G, & Lesniak, L (000) Graphs and digraphs Boca Raton, FL: CRC Press Chartrand, G, & Zhang, P (009) Chromatic graph theory Boca Raton, FL: CRC Press Effatin, B, & Kheddouci, H (003) The b-chromatic number of some power graphs Discrete Mathematics & Theoretica Computer Science, 6, Gross, J T, & Yeen, J (006) Graph theory and its appications Boca Raton, FL: CRC Press Harary, F (1969) Graph theory Phiippines: Addison Wesey Irving, R W, & Manove, D F (1999) The b-chromatic number of a graph Discrete Appied Mathematics, 91, Kok, J, Sudev, N K, & Sudev, C (in press) On the curing number of certain graphs arxiv: v Kok, J, & Sudev, N K (in press) The b-chromatic numbers of certain graphs and digraphs arxiv: Kouider, M, & Mahéo, M (00) Some bounds for the b- chromatic number of a graph Discrete Mathematics, 56, Lisna, P C, & Sunitha, M S (015) b-chromatic sum of a graph Discrete Mathematics, Agorithms and Appications, 7(3), 1 15 doi:10114/s Škrabu áková, E (in press) Thue choice number versus Thue chromatic number of graphs arxiv: v1 Vaidya, S K, & Isaac, R V (014) The b-chromatic number of some path reated graphs Internationa Journa of Computing Science and Mathematics, 4, 7 1 Vaidya, S K, & Isaac, R V (015) The b-chromatic number of some graphs Internationa Journa of Mathematics and Soft Computing, 5, Vivin, J V, & Vekatachaam, M (015) On b-chromatic number of sunet graph and whee graph famiies Journa of the Egyptian Mathematica Society, 3, 15 West, D B (001) Introduction to graph theory Dehi: Pearson Education 016 The Author(s) This open access artice is distributed under a Creative Commons Attribution (CC-BY) 40 icense You are free to: Share copy and redistribute the materia in any medium or format Adapt remix, transform, and buid upon the materia for any purpose, even commerciay The icensor cannot revoke these freedoms as ong as you foow the icense terms Under the foowing terms: Attribution You must give appropriate credit, provide a ink to the icense, and indicate if changes were made You may do so in any reasonabe manner, but not in any way that suggests the icensor endorses you or your use No additiona restrictions You may not appy ega terms or technoogica measures that egay restrict others from doing anything the icense permits Page 11 of 11