Chromatic Index of Signed Generalized Book Graphs and Signed Complete Graphs

Chromatic Index of Signed Generalized Bo ok Graphs and Signed Complete Graphs ∗ Deepak Sehra w at Departmen t of Mathematics P andit Neki Ram Sharma Go v ernmen t College Roh tak Roh tak - 124001, India Email: deepakssehra w at5@gmail.com Rohit Departmen t of Mathematics, Maharshi Da y anand Univ ersit y Roh tak Roh tak - 124001, India Email: rohit.rs24.maths@mduroh tak.ac.in Abstract A signed graph ( G, σ ) consists of a graph G and the signature σ : E ( G ) → { +1 , − 1 } . An incidence of G is a pair ( v , e ), where v is one of the end vertices of an edge e ∈ E ( G ). A prop er q -edge coloring γ of signed graph ( G, σ ) is an assignment of colors to incidences satisfying that γ ( v , e ) = − σ ( e ) γ ( w , e ) for every edge e = vw and for any t w o incidences ( v , e ) and ( v , f ), inv olving the same vertex, γ ( v , e )  = γ ( v , f ). The c hromatic index of a signed graph ( G, σ ), denoted by χ ′ ( G, σ ), is the minimum n um ber q for whic h ( G, σ ) has a prop er q -edge coloring. In this pap er, we determine the chromatic index of signed generalized b ook graphs. W e also determine the chromatic index of signed complete graphs of order up to six. Keyw ords: Chromatic index, complete graph, edge coloring, generalized b ook graph, signed graph. 2020 Mathematics Sub ject Classification: 05C15, 05C22. 1 In tro duction The main goal of this pap er is to determine the chromatic index of all signed generalized b ook graphs and all signed complete graphs of order up to six. 1 Deepak Sehra w at and Rohit All graphs considered in this pap er are finite, simple and undirected graphs. F or a graph G , V ( G ) and E ( G ) denote the vertex set and e dge set of G , resp ectiv ely . The de gr e e of a vertex v ∈ V ( G ), denoted deg G ( v ), is the num ber of edges inciden t to v . Giv en a graph G , the maximum de gr e e ∆( G ) of G is the maxim um degree among the degrees of all v ertices in G . A prop er q -e dge c oloring of a graph G is a mapping γ : E ( G ) → { 1 , . . . , q } such that adjacent edges receiv e distinct colors. The chr omatic index of a graph G , denoted by χ ′ ( G ), is the minimum n um b er q required for a prop er q -edge coloring of G . In [9], Vizing pro ved that, for every graph G , either χ ′ ( G ) = ∆( G ) or χ ′ ( G ) = ∆( G ) + 1. Moreov er, a graph G is class 1 if χ ′ ( G ) = ∆( G ) and class 2 if χ ′ ( G ) = ∆( G ) + 1. Let G be a graph and a mapping σ : E ( G ) → { +1 , − 1 } . Then a pair Σ = ( G, σ ) is called a signe d gr aph . Given a signed graph Σ, G is called the underlying gr aph and σ is called the signatur e of Σ. V ertex coloring of signed graphs w as initiated b y Zaslavsky [12]. His idea of coloring the vertices of a signed graph had compatibilit y with deletion/con traction recurrence and c hromatic p olynomials that sp ecializes to ordinary graphs when the signed graph is all p ositive. In 2016, M´ a ˇ ca jov´ a et al. [5] used the idea of color set, giv en b y Zasla vsky , to define the chromatic num ber of a signed graph and pro ved an extension of famous Brook’s theorem in the con text of signed graphs. The concept of edge coloring of signed graphs was indep enden tly prop osed by Behr [1] and Zhang et al. [13]. In this pap er, our computation is carried out along the definition of edge coloring introduced b y Behr. In [2], authors computed the chromatic index of signed generalized P etersen graph P ( n, 1) for n ≥ 5. Particularly , they prov ed that χ ′ ( P ( n, 1) , σ ) = 3 for n ≥ 5 and gav e several examples satisfying χ ′ ( P (5 , 2) , σ ) = 4 and χ ′ ( P (6 , 2) , σ ) = 4. Zheng et al. [14] considered the chromatic index of signed generalized Petersen graph P ( n, 2) and prov ed that χ ′ ( P ( n, 2) , σ ) = 3 if n ≡ 3 mo d 6( n ≥ 9) and χ ′ ( P ( n, 2) , σ ) = 4 if n = 2 p ( p ≥ 4). Recen tly , W en et al. studied the edge coloring of Cartesian product of signed graphs in [10]. Let m and n b e p ositiv e integers suc h that m ≥ 3 and n ≥ 2. The m -cycle b o ok gr aph B ( m, n, 2) consists of n copies of the cycle C m whose intersection is a path P 2 . F or m = 3, the b o ok graph B (3 , n, 2) is well-kno wn triangular b o ok graph. Shi and Song [8] obtained upp er bounds on the sp ectral radius of triangular b o ok-free graphs. Sehraw at and Bhattacharjy a [6] computed the chromatic num ber and c hromatic polynomials of all signed bo ok graphs. How ever the chromatic index of signed bo ok graphs is still unkno wn. W e generalize the family of b ook graph as follo ws. Definition 1. F or integers m ≥ 3, n ≥ 2 and k ≥ 2, the gener alize d b o ok gr aph B ( m, n, k ) consists of n copies of the cycle C m whose in tersection is a path P k . In Subsection 3.1, we will compute the c hromatic index of all signed generalized bo ok graphs. T o the best of our kno wledge, the edge coloring of signed complete graphs is also not studied an ywhere. Chromatic Index of Signed Bo ok Graphs Ho wev er, for unsigned complete graphs, it is known that the c hromatic index of a complete graph K n dep ends on the parit y of n . More precisely , χ ′ ( K n ) =      n − 1 , if n is ev en; n, if n is o dd . In Subsection 3.2, we consider the edge coloring of signed complete graphs of order up to six. W e determine the chromatic index of all signed complete graphs of order up to six. W e also show that for any signature σ , χ ′ ( K n , σ ) = n − 1 = χ ′ ( K n ), when 2 ≤ n ≤ 6 is even. It is also pro ved that for o dd v alues of n , unlike unsigned case, there are some signed complete graphs ( K n , σ ) for which χ ′ ( K n , σ ) = n − 1 < χ ′ ( K n ). The pap er ends with some conjectures and op en problems. 2 Preliminaries In this section, we present some necessary definitions, notations and results which are used to obtain our main results. In a signed graph ( G, σ ), an edge e ∈ E ( G ) is p ositive (respectively , ne gative ) if σ ( e ) = 1 (resp ectiv ely , σ ( e ) = − 1). Let σ − 1 ( − 1) = { e ∈ E ( G ) : σ ( e ) = − 1 } , then ( G, σ ) is al l-p ositive if σ − 1 ( − 1) = ∅ and al l-ne gative if σ − 1 ( − 1) = E ( G ). By ( G, +) and ( G, − ), we denote all-positive and all-negativ e signed graphs, resp ectively . A cycle is p ositive if the pro duct of its edge signs is p ositiv e and ne gative , otherwise. A signed graph is b alanc e d if its all cycles are p ositiv e, and unb alanc e d , otherwise. The terms signed graph and its balance app eared first in a pap er of F rank Harary [3]. Let v b e a v ertex of a signed graph ( G, σ ). Then Switching v in ( G, σ ) is an operation that c hanges sign of each edge inciden t to v . In general, if we switch X ⊆ V ( G ) in ( G, σ ), then we get a signed graph ( G, σ ′ ) suc h that for every edge uv ∈ E ( G ) we hav e σ ′ ( uv ) =      − σ ( uv ) , if exactly one of u, v b elongs to X, σ ( uv ) , otherwsise . Tw o signed graphs ( G, σ ′ ) and ( G, σ ) are switching e quivalent (or, simply e quivalent ) if ( G, σ ′ ) can b e obtained b y switching some of the vertices of ( G, σ ). It is denoted by ( G, σ ′ ) ∼ ( G, σ ) (or σ ′ ∼ σ if G is clear from the context). The follo wing c haracterization for tw o signed graphs to b e switching equiv alen t is giv en b y Zasla vsky . Lemma 2.1. ([11]) Two signe d gr aphs ( G, σ 1 ) and ( G, σ 2 ) ar e switching e quivalent if and only if they have the same set of ne gative cycles. The follo wing lemma is a direct consequence of Lemma 2.1. Deepak Sehra w at and Rohit Lemma 2.2. A signe d gr aph ( G, σ ) is b alanc e d if and only if it is switching e quivalent to ( G, +) . Tw o signed graphs are isomorphic to eac h other if there exists a graph isomorphism b et w een their underlying graphs preserving the edge signs. Two signed graphs are switching isomorphic to each other if one is isomorphic to a switching of other. A switching isomorphism class of G is the collection of all signed graphs, ha ving underlying graph G , in whic h any t wo signed graphs are either switc hing equiv alen t or switc hing isomorphic to each other. An incidenc e of G is a pair ( v , e ), where v is one of the end vertices of an edge e ∈ E ( G ). Th us corresp onding to every edge of G there are tw o incidences. The set of all incidences of G is denoted b y I ( G ). Behr [1] defined edge coloring of signed graphs in terms of incidences (rather than just edges themselv es) in order to incorp orate edge signs and to make the definition compatible with switching op eration. The definition of edge coloring and chromatic index of a signed graph, giv en b y Behr, is as follo ws. Definition 2. ([1]) A q -e dge c oloring γ of Σ is an assignment of colors from the set M q to each incidence of Σ sub ject to the condition that γ ( v , e ) = − σ ( e ) γ ( w , e ) for eac h edge e = vw , where M q = {± 1 , . . . , ± r } if q = 2 r and M q = { 0 , ± 1 , . . . , ± r } if q = 2 r + 1. A q -edge coloring is pr op er if for an y t w o incidences ( v , e ) and ( v, f ), inv olving the same v ertex, γ ( v , e )  = γ ( v , f ). The chr omatic index of a signed graph Σ, denoted b y χ ′ (Σ), is the minimum num ber q for which Σ has a proper q -edge coloring. F rom Definition 2, it is clear that if we assign a non-zero color c to a negative edge e , then e must receiv e the color c at b oth incidences. Ho w ever, if e is p ositiv e, then one of the incidences of e receives the color c while the other incidence receiv es the color − c . If 0 ∈ M q , then for any p ositive or negativ e edge w e can assign the color 0 to b oth of its incidences. Thus, the chromatic index of a signed graph ( G, σ ) dep ends on the underlying graph G as well as on its signature σ . Janczewski et al. [4] studied the graph G for whic h χ ′ ( G, σ ) do es not dep end on σ . T o ac hieve this goal, they in tro duced tw o new classes of graphs, namely 1 ± and 2 ± , such that graph G is class 1 ± (resp ectiv ely , 2 ± ) if and only if χ ′ ( G, σ ) = ∆( G ) (resp ectiv ely , χ ′ ( G, σ ) = ∆( G ) + 1) for all p ossible signatures σ . Since negative edges receiv e the same color at b oth of their incidences, the follo wing lemma is imme- diate. Lemma 2.3. ([1]) F or al l-ne gative signe d gr aph ( G, − ) , ther e is a one-to-one c orr esp ondenc e b etwe en pr op er q -e dge c olorings of ( G, − ) and pr op er q -e dge c olorings of G . F or an y graph G , by Lemma 2.3, it is ob vious that χ ′ ( G ) = χ ′ ( G, − ), where ( G, − ) is all-negativ e graph o v er G . Behr [1] pro v ed that the edge coloring of a signed graph is compatible with switching op eration. That is, switching do es not affect the chromatic index of a signed graph. Chromatic Index of Signed Bo ok Graphs Lemma 2.4. ([1]) Supp ose γ is a pr op er q -e dge c oloring of ( G, σ ) and supp ose ( G, σ ′ ) is obtaine d fr om ( G, σ ) by switching a vertex set X . If γ ′ is a new c oloring which is obtaine d fr om γ by ne gating al l c olors on al l incidenc es involving vertic es fr om X , then γ ′ is a pr op er q -e dge c oloring of ( G, σ ′ ) . This lemma tells us that if ( G, σ ) and ( G, σ ′ ) are switc hing equiv alen t and χ ′ ( G, σ ) = q then χ ′ ( G, σ ′ ) = q . Behr pro v ed a signed graph version of Vizing’s theorem whic h is stated as follows. Theorem 2.5. ([1]) F or a signe d gr aph ( G, σ ) , ∆( G ) ≤ χ ′ ( G, σ ) ≤ ∆( G ) + 1 . A signed graph ( G, σ ) is class 1 if χ ′ ( G, σ ) = ∆( G ) and class 2 if χ ′ ( G, σ ) = ∆( G ) + 1 . Let Σ = ( G, σ ) b e a signed graph and let γ b e a prop er q -edge coloring of Σ. The subgraph whose edge are colored using ± c with resp ect to γ is denoted by Σ c [ γ ]. If we hav e only one coloring in mind, then w e write Σ c . W e call Σ c the c-gr aph of Σ with resp ect to γ . F urthermore, the maximum degree of Σ c is t w o b ecause at most c and − c are presen t at eac h v ertex of Σ c . Therefore, every comp onent of Σ c is either a path or a cycle. If c = 0, then the maxim um degree of Σ 0 is one and so Σ 0 is a matching. In [1], author also classified signed paths and signed cycles that can b e p ossibly app ear in Σ c when c  = 0. More precisely , we hav e the following tw o results. Theorem 2.6. ([1]) Every signe d p ath c an b e pr op erly e dge c olor e d with ± a (wher e a  = 0 ). F urthermor e, every signe d p ath has exactly two differ ent ± a c olorings. Theorem 2.7. ([1]) A signe d cir cle C c an b e pr op erly c olor e d with ± a (wher e a  = 0 ) if and only if C is p ositive. F urthermor e, every p ositive cir cle has exactly two ± a c olorings. Observ ation 1. Let γ b e a prop er edge coloring of Σ. Then by Theorem 2.6, Theorem 2.7 and ab o v e discussion, it follo ws that Σ a [ γ ] consists of paths or p ositive cycles so that Σ a [ γ ] is a balanced subgraph of maxim um degree 2. W e will apply the concept of c -graph Σ c and Observ ation 1 in the computation of the chromatic index of signed complete graphs (in Subsection 3.2). A result about the chromatic index of signed cycles is the follo wing: Prop osition 2.8. ([13]) F or any signe d cycle ( C, σ ) , χ ′ ( C, σ ) =      2 , if ( C , σ ) is b alanc e d ; 3 , otherwise . 3 Main results 3.1 Signed generalized b o ok graphs Throughout this subsection, we assume that m, n, k are p ositiv e in tegers such that m ≥ 3, n ≥ 2, k ≥ 2 and m − k ≥ 1. Let the v ertex set of B ( m, n, k ) b e V ( B ( m, n, k )) = { v 1 , . . . , v k } ∪ { u i j | 1 ≤ i ≤ n, 1 ≤ Deepak Sehra w at and Rohit j ≤ m − k } , and let v 1 v 2 ...v k b e the common path to the cycles C i m , where C i m = v 1 u i 1 u i 2 ...u i m − k v k ...v 1 , for 1 ≤ i ≤ n . F or example, the cycle C 1 5 in B (5 , 3 , 3) is the cycle v 1 u 1 1 u 1 2 v 3 v 2 v 1 , where the graph B (5 , 3 , 3) is sho wn in Figure 1. v 3 v 2 v 1 u 3 2 u 3 1 u 2 2 u 2 1 u 1 2 u 1 1 Figure 1: The generalized bo ok graph B (5 , 3 , 3). Since the c hromatic index of a signed graph is inv arian t under switc hing op eration and isomorphism, so to compute the c hromatic index of all signed generalized b ook graphs it is enough to determine the c hromatic index of switching non-isomorphic signed generalized b o ok graphs. So we first determine the switc hing non-isomorphic signed generalized b o ok graphs. Theorem 3.1. L et ( B ( m, n, k ) , σ ) b e a signe d gener alize d b o ok gr aph. Then ( B ( m, n, k ) , σ ) is e quiva- lent to ( B ( m, n, k ) , τ ) , wher e τ − 1 ( − 1) ⊆ { v 1 u 1 1 , . . . , v 1 u n 1 } . F urthermor e, the numb er of switching non- isomorphic signe d B ( m, n, k ) is n + 1 . Pr o of. Let ( B ( m, n, k ) , σ ) b e a signed generalized b ook graph. It is clear that ev ery signed cycle ( C m , π ) is switching equiv alen t to a signed cycle ( C m , π ′ ), where the n umber of negative edges in ( C n , π ′ ) is at most one. Thus by suitable switc hings, if needed, we get ( B ( m, n, k ) , σ ′ ) equiv alen t to ( B ( m, n, k ) , σ ) so that ev ery negativ e edge of ( B ( m, n, k ) , σ ′ ) is inciden t to v 1 . F urther, if the edge v 1 v 2 is negativ e in ( B ( m, n, k ) , σ ′ ), switc hing v 1 will mak e it p ositiv e. Th us we get a signed generalized b ook graph ( B ( m, n, k ) , τ ) equiv alen t to ( B ( m, n, k ) , σ ), where τ − 1 ( − 1) ⊆ { v 1 u 1 1 , . . . , v 1 u n 1 } . This prov es the first part of theorem. No w let ( B ( m, n, k ) , σ ) and ( B ( m, n, k ) , τ ) b e any tw o signed generalized b o ok graphs. By part (i), without loss of generalit y , we can assume σ − 1 ( − 1) , τ − 1 ( − 1) ⊆ { v 1 u 1 1 , . . . , v 1 u n 1 } . If | σ − 1 ( − 1) | = | τ − 1 ( − 1) | , then an one-one corresp ondence b et w een σ and τ determines an isomorphism b et w een ( B ( m, n, k ) , σ ) and ( B ( m, n, k ) , τ ). If | σ − 1 ( − 1) |  = | τ − 1 ( − 1) | , then ( B ( m, n, k ) , σ ) cannot b e switch- ing isomorphic to ( B ( m, n, k ) , τ ) b ecause b oth signed graphs hav e different num b er of negative cycles C m . Therefore, the n umber of switc hing non-isomorphic signed B ( m, n, k ) is n + 1. F or each 1 ≤ l ≤ n , let σ − 1 l ( − 1) = { v 1 u 1 1 , . . . , v 1 u l 1 } . If σ − 1 0 ( − 1) = ∅ , then b y the preceding theorem, Chromatic Index of Signed Bo ok Graphs σ 0 , σ 1 , . . . , σ n are switching non-isomorphic signatures of B ( m, n, k ). This means, each ( B ( m, n, k ) , σ l ) is a represen tative of n + 1 switc hing isomorphism classes of B ( m, n, k ), where l = 0 , 1 , . . . , n . Tw o switc hing non-isomorphic signed generalized b o ok graphs ( B (5 , 3 , 3) , σ 1 ) and ( B (5 , 3 , 3) , σ 2 ) are shown in Figure 2. v 3 v 2 v 1 u 3 2 u 3 1 u 2 2 u 2 1 u 1 2 u 1 1 (a) The signed graph ( B (5 , 3 , 3) , σ 1 ). v 3 v 2 v 1 u 3 2 u 3 1 u 2 2 u 2 1 u 1 2 u 1 1 (b) The signed graph ( B (5 , 3 , 3) , σ 2 ). Figure 2: Two switching non-isomorphic signed generalized b ook graphs o ver B (5 , 3 , 3). Throughout, dashed lines represent negativ e edges and solid lines represen t positive edges. No w w e compute the v alue of χ ′ ( B ( m, n, k ) , σ l ) for l = 0 , 1 , . . . , n . Theorem 3.2. F or signatur e σ 0 , χ ′ ( B ( m, n, k ) , σ 0 ) = n + 1 . Pr o of. W e discuss tw o cases separately . Case 1. Supp ose n = 2 r for some integer r ≥ 1. In this case, we hav e that ∆( B ( m, 2 r, k )) = 2 r + 1. Th us, χ ′ ( B ( m, 2 r , k ) , σ 0 ) ≥ 2 r + 1 by Theorem 2.5. Now we give a prop er (2 r + 1)-edge coloring γ of ( B ( m, 2 r , k ) , σ 0 ) as follows (see Figure 3a for the case m = 4 , n = 2 , k = 3). 1. W e sequen tially color the incidences of the path v 1 v 2 · · · v k using the pattern (( − 1)(1)) . . . (( − 1)(1)). 2. W e sequen tially color the incidences of path v 1 u 1 1 · · · u 1 m − k using the pattern ((1)( − 1)) . . . ((1)( − 1)). F or the edge u 1 m − k v k , w e set γ ( u 1 m − k , u 1 m − k v k ) = γ ( v k , v k u 1 m − k ) = 0. 3. F or the edge v 1 u 2 1 , we set γ ( v 1 , v 1 u 2 1 ) = γ ( u 2 1 , u 2 1 v 1 ) = 0. Now w e sequentially color the incidences of the path u 2 1 · · · u 2 m − k v k using the pattern ((1)( − 1)) . . . ((1)( − 1)). 4. Finally , for i = 3 , . . . , 2 r , we sequentially color the incidences of the path v 1 u i 1 · · · u i m − k v k using the pattern (( i +1 2 )( − i +1 2 )) . . . (( i +1 2 )( − i +1 2 )) and (( − i 2 )( i 2 )) . . . (( − i 2 )( i 2 )) according as i is o dd and ev en, respectively . (This step is needed only when r ≥ 2.) The so-obtained coloring γ is clearly a prop er edge coloring of ( B ( m, 2 r, k ) , σ 0 ) for all m ≥ 3 and k ≥ 2. Case 2. Supp ose n = 2 r − 1 for some integer r ≥ 2. In this case, w e ha v e that ∆( B ( m, 2 r − 1 , k )) = 2 r . Th us, χ ′ ( B ( m, 2 r − 1 , k ) , σ 0 ) ≥ 2 r by Theorem 2.5. No w we give a prop er (2 r )-edge coloring γ of ( B ( m, 2 r − 1 , k ) , σ 0 ) as follows (see Figure 3b for the case m = 5 , n = 3 , k = 3). Deepak Sehra w at and Rohit 1. W e sequen tially color the incidences of the path v 1 v 2 · · · v k using the pattern (( − r )( r )) . . . (( − r )( r )). 2. F or i = 1 , . . . , 2 r − 1, w e sequentially color the incidences of the path v 1 u i 1 · · · u i m − k v k using the pattern (( i +1 2 )( − i +1 2 )) . . . (( i +1 2 )( − i +1 2 )) and (( − i 2 )( i 2 )) . . . (( − i 2 )( i 2 )) according as i is odd and even, resp ectiv ely . The so-obtained coloring γ is clearly a prop er edge coloring of ( B ( m, 2 r − 1 , k ) , σ 0 ) for all m ≥ 3 and k ≥ 2. F rom Cases 1 and 2, we conclude that χ ′ ( B ( m, n, k ) , σ 0 ) = n + 1 for all m ≥ 3, n ≥ 2 and k ≥ 2. 1 0 -1 1 -1 1 0 -1 1 0 0 -1 (a) Proper 3-edge coloring of ( B (4 , 2 , 3) , σ 0 ). 1 -1 2 -2 2 -2 2 -1 1 -2 -1 1 -2 1 -1 2 1 -1 2 -1 1 -2 (b) Prop er 4-edge coloring of ( B (5 , 3 , 3) , σ 0 ). Figure 3: Prop er edge colorings of all-p ositiv e graphs ov er B (4 , 2 , 3) and B (5 , 3 , 3). No w w e determine the v alue of χ ′ ( B ( m, n, k ) , σ n ) for all m ≥ 3, n ≥ 2 and k ≥ 2. By Lemma 2.3, it is obvious that χ ′ ( B ( m, n, k )) = χ ′ ( B ( m, n, k ) , − ). But it is imp ortan t to note that ( B ( m, n, k ) , − ) is switching equiv alen t to ( B ( m, n, k ) , +) or ( B ( m, n, k ) , { v 1 v 2 } ) according as m is even or o dd, resp ectiv ely . F urthermore, by switching v 1 of ( B ( m, n, k ) , { v 1 v 2 } ) we get ( B ( m, n, k ) , σ n ). These facts imply that ( B ( m, n, k ) , − ) is equiv alen t to ( B ( m, n, k ) , σ n ) for o dd v alues of m . Thus we hav e the follo wing theorem whic h directly follows from Lemma 2.3, Lemma 2.4 and Theorem 3.2. Theorem 3.3. F or any o dd m ≥ 3 , χ ′ ( B ( m, n, k ) , σ n ) = n + 1 . It remains to compute the v alue χ ′ ( B ( m, n, k ) , σ n ) for even v alues of m . W e do it in the next result. Theorem 3.4. F or any even m ≥ 4 , χ ′ ( B ( m, n, k ) , σ n ) = n + 1 . Pr o of. By switc hing v 1 in ( B ( m, n, k ) , σ n ), we get signed generalized b ook graph ( B ( m, n, k ) , { v 1 v 2 } ). Th us, due to Lemma 2.4, instead of finding the v alue of χ ′ ( B ( m, n, k ) , σ n ), we will find the v alue of χ ′ ( B ( m, n, k ) , { v 1 v 2 } ). Obviously , χ ′ ( B ( m, n, k ) , { v 1 v 2 } ) ≥ n + 1 since ∆(( B ( m, n, k )) = n + 1. W e discuss t w o cases separately . Chromatic Index of Signed Bo ok Graphs Case 1. Supp ose n = 2 r for some integer r ≥ 1. In this case, we hav e that ∆( B ( m, 2 r, k )) = 2 r + 1. Th us, χ ′ ( B ( m, 2 r , k ) , { v 1 v 2 } ) ≥ 2 r + 1 by Theorem 2.5. No w we give a prop er (2 r + 1)-edge coloring γ of ( B ( m, 2 r , k ) , { v 1 v 2 } ). Sub case 1.1. If k = 2, then γ is defined as follows (see Figure 4a for the case m = 4 , n = 2 , k = 2). 1. F or the edge v 1 v 2 , w e set γ ( v 1 , v 1 v 2 ) = γ ( v 2 , v 2 v 1 ) = 0. 2. F or i = 1 , . . . , 2 r , we sequentially color the incidences of the path v 1 u i 1 · · · u i m − k v k using the pat- tern (( i +1 2 )( − i +1 2 )) . . . (( i +1 2 )( − i +1 2 )) and (( − i 2 )( i 2 )) . . . (( − i 2 )( i 2 )) according as i is o dd and even, resp ectiv ely . Sub case 1.2. If k ≥ 3, then γ is defined as follo ws (see Figure 4b for the case m = 6 , n = 2 , k = 3). 1. F or the edge v 1 v 2 , w e set γ ( v 1 , v 1 v 2 ) = γ ( v 2 , v 2 v 1 ) = 0. 2. F or j = 2 , . . . , k − 1, incidences ( v j , v j v j +1 ) and ( v j +1 , v j +1 v j ) are colored with 1 and − 1, resp ec- tiv ely . 3. W e sequen tially color the incidences of path v 1 u 1 1 · · · u 1 m − k using the pattern ((1)( − 1)) . . . ((1)( − 1)). F or the edge u 1 m − k v k , w e set γ ( u 1 m − k , u 1 m − k v k ) = γ ( v k , v k u 1 m − k ) = 0. 4. F or i = 2 , . . . , 2 r , we sequentially color the incidences of the path v 1 u i 1 · · · u i m − k v k using the pat- tern (( − i 2 )( i 2 )) . . . (( − i 2 )( i 2 )) and (( i +1 2 )( − i +1 2 )) . . . (( i +1 2 )( − i +1 2 )) according as i is even and o dd, resp ectiv ely . The so-obtained coloring γ is clearly a proper edge coloring of ( B ( m, 2 r, k ) , { v 1 v 2 } ) for all m ≥ 3 and k ≥ 2. Case 2. Supp ose n = 2 r − 1 for some integer r ≥ 2. In this case, w e ha v e that ∆( B ( m, 2 r − 1 , k )) = 2 r . Th us, χ ′ ( B ( m, 2 r − 1 , k ) , { v 1 v 2 } ) ≥ 2 r by Theorem 2.5. No w we give a prop er (2 r )-edge coloring γ of ( B ( m, 2 r − 1 , k ) , { v 1 v 2 } ) as follows (see Figure 4c for the case m = 6 , n = 3 , k = 4). 1. F or the edge v 1 v 2 , w e set γ ( v 1 , v 1 v 2 ) = γ ( v 2 , v 2 v 1 ) = − r and if k ≥ 3, then w e color the incidences of v 2 v 3 . . . v k − 1 v k using the pattern (( r )( − r )) . . . (( r )( − r )). 2. W e sequen tially color the incidences of path v 1 u 1 1 · · · u 1 m − k using the pattern ((1)( − 1)) . . . ((1)( − 1)). F or the edge u 1 m − k v k , w e set γ ( u 1 m − k , u 1 m − k v k ) = − r and γ ( v k , v k u 1 m − k ) = r . 3. F or i = 2 , . . . , 2 r − 2, w e sequentially color the incidences of the path v 1 u i 1 · · · u i m − k v k using the pattern (( − i 2 )( i 2 )) . . . (( − i 2 )( i 2 )) and (( i +1 2 )( − i +1 2 )) . . . (( i +1 2 )( − i +1 2 )) according as i is ev en and odd, resp ectiv ely . Deepak Sehra w at and Rohit 4. W e sequentially color the incidences of the path v 1 u 2 r − 1 1 · · · u 2 r − 1 m − k using the pattern (( r )( − r )) . . . (( r )( − r )). F or the edge u 2 r − 1 m − k v k , we set γ ( u 2 r − 1 m − k , u 2 r − 1 m − k v k ) = 1 and γ ( v k , v k u 2 r − 1 m − k ) = − 1. The so-obtained coloring γ is clearly a prop er edge coloring of ( B ( m, 2 r − 1 , k ) , { v 1 v 2 } ) for all even m ≥ 4 and k ≥ 2. This completes the proof. 0 0 1 -1 1 -1 1 -1 -1 1 -1 1 -1 1 (a) Prop er 3-edge coloring of ( B (4 , 2 , 2) , { v 1 v 2 } ). 0 0 1 -1 1 -1 1 -1 1 -1 0 0 -1 1 -1 1 -1 1 -1 1 (b) Prop er 3-edge coloring of ( B (6 , 2 , 3) , { v 1 v 2 } ). -2 -2 2 -2 2 -2 1 -1 1 -1 -2 2 -1 1 -1 1 -1 1 2 -2 2 -2 1 -1 (c) Prop er 4-edge coloring of ( B (6 , 3 , 4) , { v 1 v 2 } ). Figure 4: Prop er edge colorings of some signed generalized b o ok graphs. No w, w e compute the chromatic index of χ ′ ( B ( m, n, k ) , σ 1 ), where σ − 1 1 ( − 1) = { v 1 u 1 1 } . Theorem 3.5. L et m ≥ 3 , n ≥ 2 , k ≥ 2 , then χ ′ ( B ( m, n, k ) , σ 1 ) = n + 1 . Pr o of. Recall that the set of negativ e edges, in a signed generalized b ook graph with signature σ 1 , is { v 1 u 1 1 } . W e distinguish tw o cases according to the parit y of n . Case 1. Supp ose n = 2 r for some integer r ≥ 1. Thus, χ ′ ( B ( m, 2 r , k ) , { v 1 u 1 1 } ) ≥ 2 r + 1 by Theorem 2.5. No w w e give a prop er (2 r + 1)-edge coloring γ of ( B ( m, 2 r, k ) , { v 1 u 1 1 } ) as follows (see Figure 5a for the case m = 6 , n = 2 , k = 3). 1. W e sequentially color the incidences of the path v 1 . . . v k using the pattern (( − r )( r )) . . . (( − r )( r )). 2. F or the edge v 1 u 1 1 , w e set γ ( v 1 , v 1 u 1 1 ) = γ ( u 1 1 , u 1 1 v 1 ) = 0 and we sequentially color the incidences of the path u 1 1 . . . u 1 m − k v k using the pattern (( r )( − r )) . . . (( r )( − r )). 3. F or i = 2 , . . . , 2 r − 1, w e sequentially color the incidences of the path v 1 u i 1 · · · u i m − k v k using the pattern (( i 2 )( − i 2 )) . . . (( i 2 )( − i 2 )) and (( − i − 1 2 )( i − 1 2 )) . . . (( − i − 1 2 )( i − 1 2 )) according as i is ev en and o dd, resp ectiv ely . 4. W e sequentially color the incidences of v 1 u 2 r 1 . . . u 2 r m − k using the pattern (( r )( − r )) . . . (( r )( − r )) and for the edge u 2 r m − k v k , w e set γ ( u 2 r m − k , u 2 r m − k v k ) = γ ( v k , v k u 2 r m − k ) = 0. Chromatic Index of Signed Bo ok Graphs The so-obtained coloring γ is clearly a prop er edge coloring of ( B ( m, 2 r , k ) , { v 1 u 1 1 } ) for all m ≥ 3 and k ≥ 2. Case 2. Suppose n = 2 r − 1 for some integer r ≥ 2. Th us, χ ′ ( B ( m, 2 r − 1 , k ) , { v 1 u 1 1 } ) ≥ 2 r by Theorem 2.5. Now w e give a prop er (2 r )-edge coloring γ of ( B ( m, 2 r − 1 , k ) , { v 1 u 1 1 } ) as follows (see Figure 5b for the case m = 6 , n = 3 , k = 4). 1. W e sequentially color the incidences of the path v 1 . . . v k using the pattern (( − r )( r )) . . . (( − r )( r )). 2. F or the edge v 1 u 1 1 , w e set γ ( v 1 , v 1 u 1 1 ) = γ ( u 1 1 , u 1 1 v 1 ) = 1 and we sequentially color the incidences of the path u 1 1 . . . u 1 m − k v k using the pattern (( r )( − r )) . . . (( r )( − r )). 3. F or i = 2 , . . . , 2 r − 2, w e sequentially color the incidences of the path v 1 u i 1 · · · u i m − k v k using the pattern (( − i 2 )( i 2 )) . . . (( − i 2 )( i 2 )) and (( i +1 2 )( − i +1 2 )) . . . (( i +1 2 )( − i +1 2 )) according as i is ev en and odd, resp ectiv ely . 4. W e sequentially color the incidences of v 1 u 2 r − 1 1 . . . u 2 r − 1 m − k using the pattern (( r )( − r )) . . . (( r )( − r )) and for the edge u 2 r − 1 m − k v k , w e set γ ( u 2 r − 1 m − k , u 2 r − 1 m − k v k ) = 1, γ ( v k , v k u 2 r − 1 m − k ) = − 1. The so-obtained coloring γ is clearly a proper edge coloring of ( B ( m, 2 r − 1 , k ) , { v 1 u 1 1 } ). By Cases 1 and 2 , the proof is complete. -1 1 -1 1 0 0 1 -1 1 -1 1 -1 1 -1 1 -1 1 -1 0 0 (a) Prop er 3-edge coloring of ( B (6 , 2 , 3) , σ 1 ). -2 2 -2 2 -2 2 1 1 2 -2 2 -2 -1 1 -1 1 -1 1 2 -2 2 -2 1 -1 (b) Prop er 4-edge coloring of ( B (6 , 3 , 4) , σ 1 ). Figure 5: Prop er edge colorings of some signed generalized b o ok graphs with signature σ 1 . Theorem 3.6. L et m ≥ 3 , n ≥ 3 , k ≥ 2 , then χ ′ ( B ( m, n, k ) , σ l ) = n + 1 , wher e 2 ≤ l ≤ n − 1 . Pr o of. It is clear that the set of negative edges in a signed generalized b ook graph with signature σ l is { v 1 u 1 1 , . . . , v 1 u l 1 } , where 2 ≤ l ≤ n − 1. Case 1. Supp ose n = 2 r for some in teger r ≥ 1. Sub case 1.1. Let l b e even. Th us l must lies b et w een 2 and 2 r − 2. In this case, we giv e a prop er (2 r + 1)-edge coloring γ of ( B ( m, 2 r, k ) , σ l ) as follows (see Figure 6a for the case m = 5 , n = 4 , k = 3). Deepak Sehra w at and Rohit -2 2 -2 2 0 0 1 -1 1 -1 1 1 -1 1 -1 1 -1 1 -1 1 0 0 2 -2 2 -2 2 -2 (a) Prop er 5-edge coloring of ( B (5 , 4 , 3) , σ 2 ). -2 2 -2 2 0 0 2 -2 2 -2 1 1 -1 1 -1 1 -1 -1 1 -1 1 -1 2 -2 2 -2 0 0 (b) Prop er 5-edge coloring of ( B (5 , 4 , 3) , σ 3 ). Figure 6: An illustration of Case 1 of Theorem 3.6. Prop er colorings of ( B (5 , 4 , 3) , σ 2 ) and ( B (5 , 4 , 3) , σ 3 ). 1. W e sequentially color the incidences of the path v 1 . . . v k using the pattern (( − r )( r )) . . . (( − r )( r )). 2. F or the edge v 1 u i 1 , w e set γ ( v 1 , v 1 u i 1 ) = γ ( u i 1 , u i 1 v 1 ) =            0 if i = 1 , i 2 if i is even and 2 ≤ i ≤ l, − i − 1 2 if i is o dd and 3 ≤ i ≤ l − 1 (this case o ccurs only if l ≥ 4) . 3. W e sequentially color the incidences of u 1 1 . . . u 1 m − k v k using the pattern (( l 2 )( − l 2 )) . . . (( l 2 )( − l 2 )). 4. F or i = 2 , . . . , l , we sequentially color the incidences of the path u i 1 · · · u i m − k v k using the pattern (( − i 2 )( i 2 )) . . . (( − i 2 )( i 2 )) and (( i − 1 2 )( − i − 1 2 )) . . . (( i − 1 2 )( − i − 1 2 )) according as i is ev en and odd, resp ec- tiv ely . 5. W e sequentially color the incidences of v 1 u l +1 1 . . . u l +1 m − k using the pattern (( − l 2 )( l 2 )) . . . (( − l 2 )( l 2 )) and for the edge u l +1 m − k v k , w e set γ ( u l +1 m − k , u l +1 m − k v k ) = γ ( v k , v k u l +1 m − k ) = 0. 6. F or i = l + 2 , . . . , 2 r , w e sequen tially color the incidences of the path v 1 u i 1 · · · u i m − k v k using the pattern (( i 2 )( − i 2 )) . . . (( i 2 )( − i 2 )) and (( − i − 1 2 )( i − 1 2 )) . . . (( − i − 1 2 )( i − 1 2 )) according as i is even and o dd, resp ectiv ely . The so-obtained coloring γ is clearly a proper edge coloring of ( B ( m, 2 r, k ) , σ l ), where l is even. Sub case 1.2. Let l b e o dd. Thus l must lies b etw een 3 and 2 r − 1. In this case, we give a prop er (2 r + 1)-edge coloring γ of ( B ( m, 2 r, k ) , σ l ) as follows (see Figure 6b for the case m = 5 , n = 4 , k = 3). 1. W e sequentially color the incidences of the path v 1 . . . v k using the pattern (( − r )( r )) . . . (( − r )( r )). Chromatic Index of Signed Bo ok Graphs 2. F or the edge v 1 u i 1 , w e set γ ( v 1 , v 1 u i 1 ) = γ ( u i 1 , u i 1 v 1 ) =            0 if i = 1 , i 2 if i is even and 2 ≤ i ≤ l − 1 , − i − 1 2 if i is o dd and 3 ≤ i ≤ l . 3. W e sequentially color the incidences of u 1 1 . . . u 1 m − k v k using the pattern (( r )( − r )) . . . (( r )( − r )). 4. F or i = 2 , . . . , l , we sequentially color the incidences of the path u i 1 · · · u i m − k v k using the pattern (( − i 2 )( i 2 )) . . . (( − i 2 )( i 2 )) and (( i − 1 2 )( − i − 1 2 )) . . . (( i − 1 2 )( − i − 1 2 )) according as i is ev en and odd, resp ec- tiv ely . 5. If there is an integer i such that l +1 ≤ i ≤ 2 r − 1, then we sequen tially color the incidences of the path v 1 u i 1 · · · u i m − k v k using the pattern (( i 2 )( − i 2 )) . . . (( i 2 )( − i 2 )) and (( − i − 1 2 )( i − 1 2 )) . . . (( − i − 1 2 )( i − 1 2 )) ac- cording as i is even and o dd, resp ectively . 6. W e sequentially color the incidences of v 1 u 2 r 1 . . . u 2 r m − k using the pattern (( r )( − r )) . . . (( r )( − r )) and for the edge u 2 r m − k v k , w e set γ ( u 2 r m − k , u 2 r m − k v k ) = γ ( v k , v k u 2 r m − k ) = 0. The so-obtained coloring γ is clearly a proper edge coloring of ( B ( m, 2 r, k ) , σ l ), where l is o dd. Case 2. Supp ose n = 2 r − 1 for some integer r ≥ 2. Sub case 2.1. Let l b e even. Th us l must lies b et w een 2 and 2 r − 2. In this case, we giv e a prop er (2 r )-edge coloring γ of ( B ( m, 2 r − 1 , k ) , σ l ) as follows (see Figure 7a for the case m = 5 , n = 3 , k = 3). 1. W e sequentially color the incidences of the path v 1 . . . v k using the pattern (( − r )( r )) . . . (( − r )( r )). 2. F or the edge v 1 u i 1 , w e set γ ( v 1 , v 1 u i 1 ) = γ ( u i 1 , u i 1 v 1 ) =      i +1 2 if i is o dd and 1 ≤ i ≤ l − 1 , − i 2 if i is even and 2 ≤ i ≤ l . 3. F or i = 1 , . . . , l , we sequentially color the incidences of the path u i 1 · · · u i m − k v k using the pattern (( − i +1 2 )( i +1 2 )) . . . (( − i +1 2 )( i +1 2 )) and (( i 2 )( − i 2 )) . . . (( i 2 )( − i 2 )) according as i is o dd and ev en, respec- tiv ely . 4. F or i = l + 1 , . . . , 2 r − 1, we sequentially color the incidences of the path v 1 u i 1 · · · u i m − k v k using the pattern (( i +1 2 )( − i +1 2 )) . . . (( i +1 2 )( − i +1 2 )) and (( − i 2 )( i 2 )) . . . (( − i 2 )( i 2 )) according as i is o dd and ev en, respectively . The so-obtained coloring γ is clearly a proper edge coloring of ( B ( m, 2 r − 1 , k ) , σ l ), where l is even. Sub case 2.2. Let l b e o dd. Thus l must lies b etw een 3 and 2 r − 3. In this case, we give a prop er (2 r )-edge coloring γ of ( B ( m, 2 r − 1 , k ) , σ l ) as follows (see Figure 7b for the case m = 5 , n = 5 , k = 3). Deepak Sehra w at and Rohit 1. W e sequentially color the incidences of the path v 1 . . . v k using the pattern (( − r )( r )) . . . (( − r )( r )). 2. F or the edge v 1 u i 1 , w e set γ ( v 1 , v 1 u i 1 ) = γ ( u i 1 , u i 1 v 1 ) =      i +1 2 if i is o dd and 1 ≤ i ≤ l , − i 2 if i is even and 2 ≤ i ≤ l − 1 . 3. W e color the incidences of u 1 1 . . . u 1 m − k v k using the pattern (( l +1 2 )( − l +1 2 )) . . . (( l +1 2 )( − l +1 2 )). 4. F or i = 2 , . . . , l − 1, we sequentially color the incidences of the path u i 1 · · · u i m − k v k using the pat- tern (( i 2 )( − i 2 )) . . . (( i 2 )( − i 2 )) and (( − i +1 2 )( i +1 2 )) . . . (( − i +1 2 )( i +1 2 )) according as i is even and o dd, resp ectiv ely . 5. W e sequentially color the incidences of u l 1 . . . u l m − k v k using the pattern (( − 1)(1)) . . . (( − 1)(1)). 6. F or i = l + 1 , . . . , 2 r − 1, we sequentially color the incidences of the path v 1 u i 1 · · · u i m − k v k using the pattern (( − i 2 )( i 2 )) . . . (( − i 2 )( i 2 )) and (( i +1 2 )( − i +1 2 )) . . . (( i +1 2 )( − i +1 2 )) according as i is even and o dd, resp ectiv ely . The so-obtained coloring γ is clearly a proper edge coloring of ( B ( m, 2 r − 1 , k ) , σ l ), where l is o dd. Hence the pro of follo ws from Cases 1 and 2. -2 2 -2 2 1 1 -1 1 -1 1 -1 -1 1 -1 1 -1 2 -2 2 -2 2 -2 (a) Prop er 4-edge coloring of ( B (5 , 3 , 3) , σ 2 ). -3 3 -3 3 1 1 2 -2 2 -2 -1 -1 1 -1 1 -1 2 2 -1 1 -1 1 -2 2 -2 2 -2 2 3 -3 3 -3 3 -3 (b) Prop er 5-edge coloring of ( B (5 , 5 , 3) , σ 3 ). Figure 7: An illustration of Case 2 of Theorem 3.6. Prop er colorings of ( B (5 , 3 , 3) , σ 2 ) and ( B (5 , 5 , 3) , σ 3 ). F rom Theorems 3.2 - 3.6, it follows that each signed generalized b ook graph is class 1. Chromatic Index of Signed Bo ok Graphs 3.2 Signed complete graphs In this subsection, we study edge coloring of signed complete graphs of order up to six. It is well-kno wn that there are tw o, three, seven and sixteen signed complete graphs on three, four, fiv e and six vertices, resp ectively . Their representativ es, tak en from [7], are shown in Figures 8, 9 and 10. W e determine the chromatic index of each of these represen tativ es. 1 − 1 1 − 1 1 − 1 ( K 3 , σ 1 ) 0 0 1 − 1 1 − 1 ( K 3 , σ 2 ) ( K 4 , σ 1 ) 1 − 1 1 − 1 1 − 1 1 − 1 0 0 0 0 ( K 4 , σ 2 ) 1 − 1 0 0 − 1 1 0 0 1 − 1 − 1 1 ( K 4 , σ 3 ) 1 − 1 0 0 − 1 1 0 0 1 − 1 − 1 1 ( K 5 , σ 1 ) 1 − 1 1 − 1 1 − 1 1 − 1 1 − 1 2 -2 2 -2 2 -2 2 -2 2 -2 ( K 5 , σ 2 ) 0 0 1 − 1 1 − 1 1 − 1 1 − 1 2 -2 2 -2 2 -2 2 -2 2 -2 ( K 5 , σ 3 ) 1 1 − 1 1 − 1 − 1 1 − 1 1 − 1 2 -2 2 -2 2 -2 2 -2 2 -2 ( K 5 , σ 4 ) 1 1 − 1 − 1 1 − 1 1 − 1 1 − 1 2 -2 2 -2 2 -2 2 -2 2 -2 ( K 5 , σ 5 ) 1 1 − 1 − 1 1 − 1 0 0 1 − 1 2 -2 2 -2 2 -2 2 -2 2 -2 ( K 5 , σ 6 ) 1 1 − 1 − 1 0 0 1 − 1 1 − 1 2 -2 2 -2 2 -2 2 -2 2 -2 ( K 5 , σ 7 ) 1 1 − 1 − 1 1 − 1 0 0 1 − 1 0 -2 2 0 2 -2 2 -2 2 -2 Figure 8: Switching non-isomorphic signed complete graphs of order up to fiv e and their proper edge colorings. Theorem 3.7. F or any signatur e σ , χ ′ ( K 3 , σ ) =      2 , if σ ∼ σ 1 , 3 , if σ ∼ σ 2 . Pr o of. The proof directly follows from the Prop osition 2.8. Ho wev er, signed graphs ( K 3 , σ 1 ) and ( K 3 , σ 2 ) and their prop er edge colorings are shown in Figure 8. Theorem 3.8. F or any signatur e σ , χ ′ ( K 4 , σ ) = 3 . Deepak Sehra w at and Rohit Pr o of. By Theorem 2.5, 3 ≤ χ ′ ( K 4 , σ ) ≤ 4 for every signature σ . It is known that every signed ( K 4 , σ ) is switc hing equiv alen t to one of ( K 4 , σ 1 ), ( K 4 , σ 2 ) and ( K 4 , σ 3 ) which are sho wn in Figure 8. Also a prop er 3-edge coloring of each ( K 4 , σ i ), for i = 1 , 2 , 3, is given in Figure 8. This completes the pro of. ( K 6 , σ 1 ) 1 -1 1 -1 1 -1 1 -1 1 -1 1 -1 2 0 0 -2 -2 2 0 2 2 -2 0 0 2 2 -2 -2 0 -2 ( K 6 , σ 2 ) 0 0 1 -1 1 -1 0 0 -1 1 -1 1 -2 2 -1 -2 2 -1 -2 2 0 -2 2 1 -2 2 1 2 -2 0 ( K 6 , σ 3 ) 1 1 -1 -1 1 -1 1 -1 1 -1 -1 1 2 0 0 -2 -2 2 0 2 2 -2 0 0 2 2 -2 -2 0 -2 ( K 6 , σ 4 ) 0 0 -2 2 0 0 -1 1 0 0 1 -1 1 2 -2 -1 -1 1 -2 1 1 -2 2 2 -2 2 -1 2 -2 -1 ( K 6 , σ 5 ) 2 2 0 0 -2 -2 -1 1 0 0 2 -2 -1 1 0 -1 1 -2 2 1 -1 2 -2 0 -1 -2 2 1 -1 1 ( K 6 , σ 6 ) 2 2 0 0 -2 2 -2 -2 0 0 2 -2 -1 1 0 -1 1 -2 -1 1 2 -1 1 0 -1 1 2 1 -1 -2 ( K 6 , σ 7 ) 0 0 -2 2 0 0 -1 1 0 0 1 -1 1 2 -2 -1 -1 1 -2 1 1 -2 2 2 -2 2 -1 2 -2 -1 ( K 6 , σ 8 ) 1 1 0 0 -2 2 1 -1 0 0 -2 2 -1 -2 0 2 -1 -1 2 -2 1 -1 -2 0 2 1 1 -2 2 -1 ( K 6 , σ 9 ) 1 1 -1 -1 1 1 -1 -1 1 -1 1 -1 2 0 0 -2 -2 2 0 2 2 -2 0 0 2 2 -2 -2 0 -2 Figure 9: Some switc hing non-isomorphic signed complete graphs o v er K 6 and their proper edge colorings. Prop osition 3.9. F or any signatur e σ , if Σ = ( K 5 , σ ) and χ ′ (Σ) = 4 , then Σ 1 and Σ 2 must b e p ositive cycles of length five. Pr o of. Let σ b e a signature such that χ ′ (Σ) = 4 and supp ose γ is a corresp onding prop er 4-edge coloring of Σ, where Σ = ( K 5 , σ ). If the set of colors is {± 1 , ± 2 } for coloring γ , then each comp onen t of Σ 1 [ γ ] and Σ 2 [ γ ] is either a path or a positive cycle (due to Observ ation 1). Thus the coloring γ corresp onds to a partition of the edges of Σ into balanced subgraphs of maximum degree 2. Consequently , Σ 1 and Σ 2 Chromatic Index of Signed Bo ok Graphs m ust be p ositive cycles of length 5. Theorem 3.10. F or any signatur e σ , χ ′ ( K 5 , σ ) =      4 , if σ ∼ σ i in Fig. 8 for i = 1 , 3 , 4; 5 , otherwise . ( K 6 , σ 10 ) 1 1 -1 -1 1 1 -1 1 -1 -1 1 -1 2 0 0 -2 -2 2 0 2 2 -2 0 0 2 2 -2 -2 0 -2 ( K 6 , σ 11 ) 1 1 -1 1 -1 -1 1 1 -1 1 -1 -1 2 0 0 -2 -2 2 0 2 2 -2 0 0 2 2 -2 -2 0 -2 ( K 6 , σ 12 ) 1 1 0 0 -2 2 1 -1 0 0 -2 2 -1 -2 0 2 -1 -1 2 -2 1 -1 -2 0 2 1 1 -2 2 -1 ( K 6 , σ 13 ) -1 -1 1 1 2 -2 0 0 -1 1 2 2 0 0 1 -2 0 2 -1 2 -2 1 1 -1 2 -1 -2 -2 0 -2 ( K 6 , σ 14 ) 1 1 -1 -1 1 -1 0 0 -1 -1 -2 2 0 0 -2 2 0 -2 2 -2 -2 1 -2 2 1 1 2 -1 0 2 ( K 6 , σ 15 ) 1 1 -1 -1 1 1 -1 -1 1 -1 1 -1 2 2 -2 0 -2 -2 2 0 0 -2 -2 2 0 2 2 0 -2 0 ( K 6 , σ 16 ) 2 2 -2 -2 2 -2 2 2 -2 -2 2 -2 0 -1 -1 1 0 0 -1 -1 1 0 1 1 -1 0 0 1 1 -1 Figure 10: A contin uation of Figure 9. Remaining switching non-isomorphic signed complete graphs ov er K 6 (not sho wn in Figure 9) and their proper edge colorings. Pr o of. By Theorem 2.5, 4 ≤ χ ′ ( K 5 , σ ) ≤ 5 for every signature σ . Therefore if σ ∼ σ i for i = 1 , 3 , 4, then to prov e χ ′ ( K 5 , σ ) = 4 , it suffices to giv e a prop er 4-coloring of ( K 5 , σ ). Such a prop er 4-coloring of ( K 5 , σ i ) is given in Figure 8 for eac h i = 1 , 3 , 4. Hence χ ′ ( K 5 , σ ) = 4 for σ ∼ σ i , where i = 1 , 3 , 4. F or σ ∼ σ 2 , if χ ′ ( K 5 , σ 2 ) = 4, then by Prop osition 3.9, Σ 1 and Σ 2 m ust b e p ositiv e cycles of length 5. But ( K 5 , σ 2 ) has exactly one negative edge, namely , u 1 u 2 and therefore either Σ 1 or Σ 2 has to consist Deepak Sehra w at and Rohit u 1 u 2 . Consequen tly , either Σ 1 or Σ 2 is a negative cycle, a contradiction. This pro v es that χ ′ ( K 5 , σ 2 ) = 5. Similarly , for j = 5 , 6, if σ ∼ σ j and χ ′ ( K 5 , σ ) = 4, then b y Prop osition 3.9, Σ 1 and Σ 2 m ust be p ositiv e cycles of length 5. But this is not p ossible because signed graphs ( K 5 , σ 5 ) and ( K 5 , σ 6 ) hav e exactly 3 negative edges making either Σ 1 or Σ 2 negativ e. Consequently χ ′ ( K 5 , σ j ) = 5 for j = 5 , 6. In ( K 5 , σ 7 ), it is easy to v erify that ev ery C 5 is negative. Th us due to Proposition 3.9, the chromatic index of ( K 5 , σ 7 ) cannot b e 4. Hence χ ′ ( K 5 , σ 7 ) = 5. This completes the proof. Theorem 3.11. F or any signatur e σ , χ ′ ( K 6 , σ ) = 5 . Pr o of. By Theorem 2.5, 5 ≤ χ ′ ( K 6 , σ ) ≤ 6 for every signature σ . It is clear that ev ery signed ( K 6 , σ ) is switc hing equiv alent to one of the signed complete graphs given in Figures 9 and 10. Thus, to complete the proof, it is sufficien t to giv e a prop er 5-edge coloring of each ( K 6 , σ i ) for i = 1 , . . . , 16 . A prop er 5-edge coloring of each ( K 6 , σ i ) is given in Figures 9 and 10. This completes the proof. In this subsection, we studied edge coloring of signed complete graphs of order up to 6. F urther, we observ ed that the chromatic index of a signed complete graph ( K n , σ ) dep ends on the v alue of n as well as signature σ . How ever, in case of even v alues of n (here n = 4 , 6), the chromatic index do es not dep end on the signature. 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