The Undirected Optical Indices of Trees
TTHE UNDIRECTED OPTICAL INDICES OF TREES
YUAN-HSUN LO, HUNG-LIN FU, YIJIN ZHANG, AND WING SHING WONG
Abstract.
For a connected graph G , an instance I is a set of pairs of vertices and acorresponding routing R is a set of paths specified for all vertex-pairs in I . Let R I bethe collection of all routings with respect to I . The undirected optical index of G withrespect to I refers to the minimum integer k to guarantee the existence of a mapping φ : R → { , , . . . , k } , such that φ ( P ) (cid:54) = φ ( P (cid:48) ) if P and P (cid:48) have common edge(s), overall routings R ∈ R I . A natural lower bound of the undirected optical index is the edge-forwarding index, which is defined to be the minimum of the maximum edge-load over allpossible routings. Let w ( G, I ) and π ( G, I ) denote the undirected optical index and edge-forwarding index with respect to I , respectively. In this paper, we derive the inequality w ( T, I A ) < π ( T, I A ) for any tree T , where I A := {{ x, y } : x, y ∈ V ( T ) } is the all-to-allinstance. Introduction
Let G be a connected graph with vertex set V ( G ) and edge set E ( G ). An instance I is aset (or, multiset) of vertex-pairs of V ( G ). A routing R in G with respect to I is a set of | I | paths, one for each vertex-pair in I . That is, { x, y } ∈ I if and only if there is a path having x and y as its terminal vertices. Such a path is denoted by P x,y or P y,x . A k -path-coloring of R is a mapping φ : R → { , , . . . , k } , and is said to be proper if φ ( P ) (cid:54) = φ ( P (cid:48) ) whenever P and P (cid:48) have common edge(s). Let w ( G, I, R ) be the minimum integer k to guarantee theexistence of a proper k -path-coloring of R . Let R I denote the collection of all routings in G with respect to I . The undirected optical index (or the path-chromatic number ) of G withrespect to I is then defined to be w ( G, I ) := min R ∈ R I w ( G, I, R ) . Note that by constructing a graph Q ( R ), say conflict graph , on R by paths P and P (cid:48) beingadjacent if and only if they have common edge(s), the value w ( G, I, R ) turns out to be thechromatic number of Q ( R ), i.e., χ ( Q ( R )).For a routing R ∈ R I and an edge e ∈ E ( G ), the edge-load of e , denoted by (cid:96) G,R ( e ), isthe number of paths in R passing through e . Let π ( G, I, R ) denote the maximum value of
Mathematics Subject Classification.
Key words and phrases. optical index; forwarding index; path-coloring; all-to-all routing.This work was supported in part by the National Natural Science Foundation of China under grantnumbers 61301107 and 11601454, the Natural Science Foundation of Fujian Province of China under grantnumber 2016J05021, the Fundamental Research Funds for the Central Universities in China under grantnumbers 20720150210 and 30918011318, the Ministry of Science and Technology, Taiwan under grant number104-2115-M-009-009, and the open research fund of National Mobile Communications Research Laboratory,Southeast University, under grant number 2017D09. a r X i v : . [ m a t h . C O ] A ug YUAN-HSUN LO, HUNG-LIN FU, YIJIN ZHANG, AND WING SHING WONG (cid:96)
G,R ( e ) by going through all edges in G , i.e., π ( G, I, R ) = max e ∈ E ( G ) (cid:96) G,R ( e ). The edge-forwarding index of G with respect to I is then defined by π ( G, I ) := min R ∈ R I π ( G, I, R ) . It is easy to see that π ( G, I ) ≤ w ( G, I ) for any connected graph G and instance I .Analogous parameters can be introduced when considering a connected bidirected graph,which is a digraph obtained from a connected (undirected) graph by putting two oppositearcs on each edge. In a bidirected graph G , an instance I consists of ordered pairs of verticesand a corresponding routing (cid:126)R I refers to a set of | I | dipaths specified for all ordered pairs in I . The optical index and arc-forwarding index , denoted by (cid:126)w ( G, I ) and (cid:126)π ( G, I ) respectively,are defined accordingly. We use the right-arrow symbol to emphasize that the parametersare considered in a directed version. It is worth noting that the evaluation of opticalindices is known as the routing and wavelength assignment (RWA) problem, which arisesfrom the investigation of optimal wavelength allocation in an optical network that employsWavelength Division Multiplexing (WDM) [1].For an arbitrary instance I , to evaluate the exact value of w ( G, I ) has been shownto be NP-hard, even for trees [13] and cycles [8]. Some approximation algorithms wereproposed in [23, 20]. The best known results are with approximation ratio for trees [8]and approximation ratio 2 − o (1) for cycles [7]. When it comes to directed case, it is alsoNP-hard to determine (cid:126)w ( G, I ) for trees and cycles [8]. A -approximation algorithm fortrees was proposed in [9] and a 2-approximation algorithm for cycles was given in [6]. As (cid:126)π ( G, I ) being a natural lower bound of (cid:126)w ( G, I ), Kaklamanis et al. [16] showed that (cid:126)π ( G, I )colors are enough when G is a tree, and Tucker [24] showed that 2 (cid:126)π ( G, I ) − G is a cycle. Interested readers are referred to [4, 12, 14, 17, 19, 21] for moreinformation.Some literatures focused on the fundamental case when the instance consists of all vertex-pairs (or, ordered pairs of vertices for directed case), called all-to-all instance and denotedby I A . That is, | I A | = (cid:0) | V ( G ) | (cid:1) for undirected case and | I A | = | V ( G ) | ( | V ( G ) | −
1) fordirected case. It has been proved that the equality (cid:126)w ( G, I A ) = (cid:126)π ( G, I A ) holds for trees [11],cycles [25], trees of cycles [5], some Cartesian product of paths or cycles with equal lengths [3,22], some certain compound graphs [2] and circulant graphs [2, 12]. Kosowski [15] provideda family of graphs satisfying (cid:126)w ( G, I A ) > (cid:126)π ( G, I A ).The results for all-to-all instance on undirected case are relatively few. The exact valueof w ( G, I A ) and the gap between it to π ( G, I A ) are characterized for cycles [18] or complete m -ary trees [10]. It was conjectured in [10] that w ( G, I A ) is upper bounded by π ( G, I A )in the case when G is a tree. This paper is devoted to prove this conjecture. It shouldbe noted here that, both the -approximation algorithm in [8] for undirected case and themethod of the usage of (cid:126)π ( G, I ) colors in [16] for directed case do not cover our result.2.
Main Result
Let T be a tree. There is a unique path to connect any pair of vertices in T , so | R I | = 1,for any instance I . Hereafter we only consider the all-to-all instance and use R to denote HE UNDIRECTED OPTICAL INDICES OF TREES 3 the unique all-to-all routing. For convenience, w ( T, I A ), π ( T, I A ) and (cid:96) T,R ( e ) are simplywritten as w ( T ), π ( T ) and (cid:96) T ( e ), respectively.Since each edge e ∈ E ( T ) is a bridge, (cid:96) T ( e ) is equal to the product of the numbers ofvertices of the two components in T − e . Therefore, a natural upper bound of π ( T ) isobtained as follows. Proposition 1.
Let T be a tree of order n . It follows that π ( T ) ≤ n . Here is the main result in this paper.
Theorem 2.
Let T be a tree of order n . It follows that w ( T ) < π ( T ) . (1) Proof.
The proof of Theorem 2 is obtained by induction on n . It is obvious that (1) holdswhen n ≤
3. In what follows we consider n ≥
4, and assume (1) holds for any tree of orderless than n .Let ˆ e be the edge maximizing the value (cid:96) T ( e ) among all edges, that is, (cid:96) T (ˆ e ) = π ( T ).Note that ˆ e may not be unique. Let A and B be the two connected components of T − ˆ e with a ≥ b , where a := | V ( A ) | and b := | V ( B ) | .We first consider the case when b ≥ a . The paths in R are partitioned into three classes. • P := { P x,y : x, y ∈ V ( A ) } . • P := { P x,y : x, y ∈ V ( B ) } . • P := { P x,y : x ∈ V ( A ) and y ∈ V ( B ) } .It follows from previous observation that χ ( Q ( P )) = w ( A ), χ ( Q ( P )) = w ( B ) and χ ( Q ( P )) = |P | = a · b . Observe that any two paths, one in P and another in P , can receive the samecolor. By the induction hypothesis and Proposition 1, it follows that w ( T ) ≤ max { w ( A ) , w ( B ) } + |P | < max (cid:8) π ( A ) , π ( B ) (cid:9) + |P | = max (cid:8) a , b (cid:9) + ab = a (cid:0) a + b (cid:1) ≤ a (cid:0) b + b (cid:1) = 32 ab = 32 π ( T ) , as desired.In what follows, consider b < a . Denote by r the endpoint of ˆ e with r ∈ V ( A ). Let B (= B ) , B , . . . , B d be the connected components of T − r , where d = deg T ( r ). Notethat d ≥ n ≥
4. For i = 1 , , . . . , d denote by s i the neighbor of r with s i ∈ V ( B i ), and let b i = | V ( B i ) | . The assumption π T (ˆ e ) = π ( T ) implies that b ≥ b i for i ≥
2. Without loss of generality, we assume b ≥ b ≥ · · · ≥ b d . See Fig. 1 forthe illustration of the structure of T . Notice that b = b ≤ a = 1 + b + · · · + b d and π ( T ) = ab = b (1 + b + · · · + b d ). YUAN-HSUN LO, HUNG-LIN FU, YIJIN ZHANG, AND WING SHING WONG 𝑟𝑟 ⋯ ⋯ 𝑠𝑠 𝑠𝑠 𝑠𝑠 𝑠𝑠 𝑑𝑑 𝐵𝐵 = 𝐵𝐵 𝐵𝐵 𝐵𝐵 𝐵𝐵 𝑑𝑑 𝐴𝐴̂𝑒𝑒
Figure 1.
The structure of T .To render the paper more readable, the rest of the proof is moved to the next two sections.The framework of the whole proof is illustrated as follows. b ≥ a (Section 2) b < a d = 2 (Section 3.1) d = 3 (Section 3.2) d = 4 (Section 3.3) d ≥ b ≥ a/ b < a/ (cid:40) b = b = · · · = b d (Section 4.2) b = · · · = b k > b k +1 ≥ · · · ≥ b d (Section 4.3) (cid:3) Proof of Theorem 2 for d < a and ≤ d ≤ d = Let e be the edge connecting r and s . Then, π ( e ) = ( b + 1)( a − > ab = π ( T )whenever 3 a > b . This contradicts to the definition of π ( T ).3.2. d = In this case, a = 1 + b + b and π ( T ) = b (1 + b + b ). The paths in R areclassified into the following 7 classes. • P := { P r,y : y ∈ V ( T ) \ { r }} . • P := { P x,y : x ∈ V ( B ) , y ∈ V ( B ) } . • P := { P x,y : x ∈ V ( B ) , y ∈ V ( B ) } . • P := { P x,y : x ∈ V ( B ) , y ∈ V ( B ) } . • P := { P x,y : x, y ∈ V ( B ) } . • P := { P x,y : x, y ∈ V ( B ) } . • P := { P x,y : x, y ∈ V ( B ) } . HE UNDIRECTED OPTICAL INDICES OF TREES 5
First, two paths in P can receive the same color if their terminal vertices (except for theone r ) are not in the same set V ( B i ), i = 1 , b ≥ b ≥ b , one has χ ( Q ( P )) ≤ b . Second, any two paths, one in P and the other in P , have no edges in common. Bythe fact χ ( Q ( P )) = |P | = b b , the induction hypothesis that χ ( Q ( P )) < π ( B ) and π ( B ) ≤ b from Proposition 1, we have χ ( Q ( P ∪P )) ≤ max { b b , b } = b b . Similarly, χ ( Q ( P ∪ P )) ≤ max { b b , b } and χ ( Q ( P ∪ P )) ≤ max { b b , b } . It follows that w ( T ) ≤ b + b b + max (cid:8) b b , b (cid:9) + max (cid:8) b b , b (cid:9) . (2)Consider the case when b b ≥ b . Since b b ≥ b due to b ≥ b ≥ b , it follows from(2) that w ( T ) ≤ b + b b + b b + b b = b (1 + b + b ) + 12 b b + 12 b b ≤ b (1 + b + b ) + 12 b b + 12 b b < b (1 + b + b ) + 12 b (1 + b + b )= 32 π ( T ) . Next, consider the case when b b < b . If b b ≥ b , by (2) and the assumption that b < a we have w ( T ) ≤ b + b b + b b + 38 b < b (1 + b + b ) + 932 b (1 + b + b ) < b (1 + b + b ) = 32 π ( T ) . Otherwise, by (2) and b < a again we have w ( T ) ≤ b + b b + 38 b + 38 b ≤ b + b b + 34 b < b + b b + 919 b (1 + b + b ) = 32 b (1 + b + b ) + 116 b (1 + b − b ) . (3)Notice that b < a and b ≤ b imply b ≤ b . It follows from (3) that w ( T ) < b (1 + b + b ) − b (3 b − < b (1 + b + b ) = 32 π ( T ) . d = In this case, a = 1 + b + b + b and π ( T ) = b (1 + b + b + b ). We considerthe following sub-cases.(i) b ≥ b + b .(ii) b < b + b and b b ≥ b b .(iii) b < b + b , b b < b b and b ≤ b .(iv) b < b + b , b b < b b and b > b .3.3.1. Proof for sub-case (i) . We first obtain a new graph T (cid:48) from T by removing theedge { r, s } and adding the edge { s , s } , see Fig. 2 for an example of T (cid:48) . It is not hard tosee that the edge ˆ e still maximizes the value (cid:96) T (cid:48) ( e ), which implies that π ( T (cid:48) ) = b (1 + b + YUAN-HSUN LO, HUNG-LIN FU, YIJIN ZHANG, AND WING SHING WONG b + b ) = π ( T ). Observe that deg T (cid:48) ( r ) = 3. By the same argument in Section 3.2 for d = 3case, we have w ( T (cid:48) ) < π ( T (cid:48) ) = 32 π ( T ) . (4) 𝑟𝑟𝑠𝑠 𝑠𝑠 𝑠𝑠 𝑠𝑠 ̂𝑒𝑒 𝑒𝑒 𝑒𝑒 𝑟𝑟𝑠𝑠 𝑠𝑠 𝑠𝑠 𝑠𝑠 ̂𝑒𝑒 𝑒𝑒𝑒 𝑇𝑇 𝑇𝑇𝑒 𝑒𝑒 Figure 2. T (cid:48) = T − { r, s } + { s , s } .Let φ (cid:48) be a proper w ( T (cid:48) )-path-coloring of T (cid:48) . Define a path-coloring φ of T by φ ( P x,y ) = φ (cid:48) ( P (cid:48) s ,y ) , if x = r and y ∈ V ( B ); (5a) φ (cid:48) ( P (cid:48) r,y ) , if x = s and y ∈ V ( B ); (5b) φ (cid:48) ( P (cid:48) x,y ) , otherwise. (5c)We use the superscript “prime” herein to emphasize the paths are considered in T (cid:48) . φ is well-defined since it just exchanges the color of the path connecting r and y with thatconnecting s and y , for any y ∈ V ( B ). Lemma 3.
The path-coloring defined on T in (5a) – (5c) is proper.Proof. Following the definition of φ , we first define a mapping f from R to the routing of T (cid:48) as f ( P x,y ) = P (cid:48) s ,y , if x = r and y ∈ V ( B ); P (cid:48) r,y , if x = s and y ∈ V ( B ); P (cid:48) x,y , otherwise.As such, φ ( P x,y ) = φ (cid:48) ( f ( P x,y )). For convenience, use e , e and e (cid:48) to denote the edges { r, s } , { r, s } and { s , s } , respectively, as shown in Fig. 2.Consider two paths in T , say P x,y and P u,v , which have at least one common edge. Let E (resp., E (cid:48) ) denote the collection of common edges of P x,y and P u,v (resp., f ( P x,y ) and f ( P u,v )). If E (cid:48) (cid:54) = ∅ , then φ ( P x,y ) = φ (cid:48) ( f ( P x,y )) (cid:54) = φ (cid:48) ( f ( P u,v )) = φ ( P u,v ) . In what follows, we aim to prove that E (cid:48) (cid:54) = ∅ .When f ( P x,y ) = P x,y and f ( P u,v ) = P u,v , one has E (cid:48) = E (cid:54) = ∅ . Consider the case when f ( P x,y ) (cid:54) = P x,y and f ( P u,v ) (cid:54) = P u,v . By assuming y, v ∈ V ( B ), there are four possibilitiesfor the choices of x, u : x = u = r ; x = u = s ; x = r and u = s ; and x = s and u = r .For either case, one can check that e (cid:48) ∈ E (cid:48) . Hence E (cid:48) (cid:54) = ∅ . HE UNDIRECTED OPTICAL INDICES OF TREES 7
Finally, we consider by symmetry that f ( P x,y ) (cid:54) = P x,y and f ( P u,v ) = P u,v . By assuming y ∈ V ( B ), one has x = r or x = s . When x = r (i.e., f ( P x,y ) = P (cid:48) s ,y ), E (cid:54) = ∅ and f ( P u,v ) = P u,v imply that either u, v ∈ V ( B ) or u ∈ V ( B ) ∪ V ( B ) ∪ V ( B ) and v ∈ V ( B ).For the former case we have E (cid:48) = E , while for the latter case we have e (cid:48) ∈ E (cid:48) . When x = s (i.e., f ( P x,y ) = P (cid:48) r,y ), by E (cid:54) = ∅ and f ( P u,v ) = P u,v again, there are three possibilities forthe choices of u, v : (i) u, v ∈ V ( B ); (ii) u ∈ V ( B ) ∪ V ( B ) ∪ V ( B ) and v ∈ V ( B ); and(iii) u ∈ V ( B ) ∪ V ( B ) and v ∈ V ( B ). We have E (cid:48) = E for (i), e (cid:48) ∈ E (cid:48) for (ii), and e ∈ E (cid:48) for (iii). This concludes that E (cid:48) (cid:54) = ∅ . (cid:3) Lemma 3 guarantees that w ( T ) ≤ w ( T (cid:48) ). Hence the result follows by (4).3.3.2. Proof for sub-cases (ii) and (iii) . We consider (ii) and (iii) simultaneously. Thepaths in R are classified into the following 5 classes. • P := { P r,y : y ∈ V ( T ) \ { r }} . • P := { P x,y : x, y ∈ V ( B i ) for some i, ≤ i ≤ } . • P := { P x,y : x ∈ V ( B ) and y ∈ V ( B ) or x ∈ V ( B ) and y ∈ V ( B ) } . • P := { P x,y : x ∈ V ( B ) and y ∈ V ( B ) or x ∈ V ( B ) and y ∈ V ( B ) } . • P := { P x,y : x ∈ V ( B ) and y ∈ V ( B ) or x ∈ V ( B ) and y ∈ V ( B ) } .Similar to the argument in (2) of Section 3.2 for d = 3 case, we have w ( T ) ≤ b + max (cid:8) b , b , b , b (cid:9) + max { b b , b b } + max { b b , b b } + max { b b , b b } . By the assumption that b ≥ b ≥ b ≥ b , the above inequality can be simplified as w ( T ) ≤ b + 38 b + b b + b b + max { b b , b b } . (6)Consider (ii): b < b + b and b b ≥ b b . It follows from (6) and b < a that w ( T ) ≤ b + 38 b + b b + b b + b b < b (1 + b + b + b ) + 38 b ·
34 (1 + b + b + b )= 4132 b (1 + b + b + b ) < b (1 + b + b + b )= 32 π ( T ) . YUAN-HSUN LO, HUNG-LIN FU, YIJIN ZHANG, AND WING SHING WONG
Consider (iii): b < b + b , b b < b b and b ≤ b . It follows from (6) that w ( T ) ≤ b + 38 b + b b + b b + b b = 32 b (1 + b + b + b ) + 38 b + b b − b − b b − b b − b b (7) ≤ b (1 + b + b + b ) + 38 b ( b − b −
43 ) (8) < b (1 + b + b + b ) (9)= 32 π ( T ) , where inequalities (8) and (9) are due to 2 b b ≤ b b + b b and b ≤ b , respectively.3.3.3. Proof for sub-case (iv) . Finally, consider (iv): b < b + b , b b < b b and b > b . The paths in R are classified into the following 7 classes. • P := { P r,y : y ∈ V ( T ) \ { r }} . • P := { P x,y : x ∈ V ( B ) and y ∈ V ( B ) } . • P := { P x,y : x, y ∈ V ( B ) ∪ V ( B ) } . • P := { P x,y : x ∈ V ( B ) and y ∈ V ( B ) } . • P := { P x,y : x, y ∈ V ( B ) ∪ V ( B ) } . • P := { P x,y : x ∈ V ( B ) and y ∈ V ( B ) } . • P := { P x,y : x, y ∈ V ( B ) ∪ V ( B ) } .It is easy to see that χ ( Q ( P )) = b , χ ( Q ( P )) = b b , χ ( Q ( P )) = b b , and χ ( Q ( P )) = b b . Let (cid:98) B be the tree obtained from the union of B and B by adding an extra edgeconnecting s and s . It is easy to see that w ( (cid:98) B ) = χ ( Q ( P )). By the induction hypothesisthat w ( (cid:98) B ) < π ( (cid:98) B ), it follows from Proposition 1 that χ ( Q ( P )) = w ( (cid:98) B ) < π ( (cid:98) B ) ≤
38 ( b + b ) . Furthermore, since any two paths, one in P and another in P , can receive the same color,we have χ ( Q ( P ∪ P )) = max (cid:8) χ ( Q ( P )) , χ ( Q ( P )) (cid:9) ≤ max (cid:8) b b ,
38 ( b + b ) (cid:9) . (10)By the same argument, one has χ ( Q ( P ∪ P )) ≤ max (cid:8) b b ,
38 ( b + b ) (cid:9) (11)and χ ( Q ( P ∪ P )) ≤ max (cid:8) b b ,
38 ( b + b ) (cid:9) . (12) HE UNDIRECTED OPTICAL INDICES OF TREES 9
Combining χ ( Q ( P )) = b and equations (10)–(12) yields w ( T ) ≤ χ ( Q ( P )) + (cid:88) i =1 χ ( Q ( P i ∪ P i +1 )) ≤ b + max (cid:8) b b ,
38 ( b + b ) (cid:9) + max (cid:8) b b ,
38 ( b + b ) (cid:9) + max (cid:8) b b ,
38 ( b + b ) (cid:9) . (13)As b < b + b and b > b implying b < b and b < b , by b ≤ b ≤ b ≤ b wehave 38 ( b + b ) <
38 ( b + 14 b )( b + b ) = 1516 b b , (14)and 38 ( b + b ) <
38 ( b + 14 b ) = 75128 b < b · b = 2532 b b . (15)By plugging (14) and(15) into (13), we have w ( T ) ≤ b + b b + b b + max (cid:8) b b ,
38 ( b + b ) (cid:9) . (16)If b b ≥ ( b + b ) , it follows (16) that w ( T ) ≤ b + b b + b b + b b = 32 b (cid:0) b + b + b (cid:1) + (cid:0) b b − b ( b + b ) (cid:1) − b − b b ≤ π ( T ) − b − b b < π ( T ) . Else, if b b < ( b + b ) , it follows (16) again that w ( T ) ≤ b + b b + b b + 38 ( b + b ) = 32 b (1 + b + b + b ) − b − b b − b b − b b + 38 b + 38 b = 32 π ( T ) − b − b b + 38 b (cid:0) b − b (cid:1) + 38 b (cid:0) b − b (cid:1) < π ( T ) − b − b b (17) < π ( T ) , where (17) is due to b < b and b ≤ b . Proof of Theorem 2 for b < a and d ≥ d , with initial cases d = 2 , b ≥ a/
2; (ii) b < a/ b = b = · · · = b d ; and (iii) b < a/ b = b = · · · = b k > b k +1 ≥ · · · ≥ b d for some k with 1 ≤ k ≤ d −
1. In each case, weassume b < b d − + b d , otherwise, it can be reduced to the d − b ≥ a / Since d ≥
5, it follows that 2 (cid:0) b d − + b d (cid:1) ≤ b + b + · · · + b d ≤ b , whichimplies that b d − + b d ≤ b . This is a contradiction to the assumption that b < b d − + b d .4.2. b < a / b = b = · · · = b d . In this case, π ( T ) = b ( bd − b + 1). Consider thefollowing classification of R . • P i := (cid:8) P x,y : x, y ∈ V ( B i ) ∪ { r } (cid:9) , for i = 1 , , . . . , d ; and • P ( i,j ) := { P x,y : x ∈ V ( B i ) and y ∈ V ( B j ) } , for 1 ≤ i < j ≤ d .First, assume d is even. For i (cid:54) = i (cid:48) , any two paths, one in P i and another in P i (cid:48) , canreceive the same color. Then, by the induction hypothesis and Proposition 1 we have χ (cid:32) Q (cid:32) d (cid:91) i =1 P i (cid:33)(cid:33) = χ ( Q ( P )) = w ( B ) < π ( B ) ≤ b . Recall that the chromatic index of a complete graph of order d is d − d is even. Let K d be a complete graph of d vertices labelled 1 , , . . . , d , and let f : E ( K d ) → { , , . . . , d − } be a proper ( d − K d . For t = 1 , , . . . , d −
1, denote by C t the collectionof ordered pairs ( i, j ) such that f ( { i, j } ) = t , where i < j . Any two paths, one in P ( i,j ) andanother in P ( i (cid:48) ,j (cid:48) ) , can receive the same color if ( i, j ) and ( i (cid:48) , j (cid:48) ) are distinct and both in C t for some t . This implies that, for any t , χ Q (cid:91) ( i,j ) ∈ C t P ( i,j ) = χ (cid:0) Q ( P ( i,j ) ) (cid:1) = | P ( i,j ) | = b . To sum up, we have w ( T ) ≤ χ (cid:32) Q (cid:32) d (cid:91) i =1 P i (cid:33)(cid:33) + d − (cid:88) t =1 χ Q (cid:91) ( i,j ) ∈ C t P ( i,j ) < b + ( d − d < b ( bd − b + 1) = 32 π ( T ) . Second, assume d is odd. Recall that the total-chromatic number of a graph G is theminimum integer k needed to guarantee the existence of a mapping from V ( G ) ∪ E ( G ) toa set of k colors such that (i) adjacent vertices receive distinct colors, (ii) incident edgesreceive distinct colors, and (iii) any vertex and its incident edges receive distinct colors.The total-chromatic number of K d is known to be d when d is odd, see [27, p.16].For convenience, label the set of vertices in K d by 1 , , . . . , d . Let f : V ( K d ) ∪ E ( K d ) →{ , , . . . , d } be a proper d -total-coloring of K d such that f ( t ) = t for any t ∈ V ( K d ). By a HE UNDIRECTED OPTICAL INDICES OF TREES 11 similar argument, for t = 1 , , . . . , d , one has χ Q P t ∪ (cid:91) f ( { i,j } )= t P ( i,j ) = max (cid:110) b , b (cid:111) = b . Therefore, w ( T ) ≤ d (cid:88) t =1 χ Q P t ∪ (cid:91) f ( { i,j } )= t P ( i,j ) = d · b < b ( bd − b + 1) = 32 π ( T ) . b < a / b = b = · · · = b k > b k + for some k with 1 ≤ k ≤ d − Constructa tree T (cid:48) from T by removing a leave u i from B i , where u i is arbitrarily chosen but vertex s i , for i = 1 , , . . . , k . Let B (cid:48) i = B i − u i for i = 1 , , . . . , k . See Fig. 3 for an example of T and T (cid:48) with d = 5 and k = 3. 𝑟𝑟𝑠𝑠 𝑠𝑠 𝑠𝑠 𝑠𝑠 ̂𝑒𝑒 𝑇𝑇 𝑠𝑠 𝑢𝑢 𝑢𝑢 𝑢𝑢 𝑟𝑟𝑠𝑠 𝑠𝑠 𝑠𝑠 𝑠𝑠 ̂𝑒𝑒 𝑇𝑇𝑒 𝑠𝑠 Figure 3. T (cid:48) = T − u − u − u , where u i is an arbitrarily chosen leaf for i = 1 , , e still maximizes the value (cid:96) T (cid:48) ( e ) among all edges in T (cid:48) , that is, (cid:96) T (cid:48) (ˆ e ) = π ( T (cid:48) ). Note here that T (cid:48) − r has d branches B (cid:48) , . . . , B (cid:48) k , B k +1 , . . . , B d . By theinduction hypothesis, we have w ( T (cid:48) ) < π ( T (cid:48) ) = 32 ( b − (cid:0) b −
1) + · · · + ( b k −
1) + b k +1 + · · · + b d (cid:1) (18)= 32 b (1 + b + · · · + b d ) − (cid:0) ( k − b + b + b + · · · + b d − ( k − (cid:1) (19)= 32 π ( T ) − k − b −
32 ( b k +1 + b k +2 + · · · + b d ) + 32 ( k − , (20)where the last equation is due to b = b = · · · = b k . Let R (cid:48) be the all-to-all routing of T (cid:48) .As w ( T ) ≤ w ( T (cid:48) ) + χ ( Q ( R \ R (cid:48) )), we consider the remaining paths in R \ R (cid:48) . When k = 1, R \ R (cid:48) = (cid:8) P u ,y : y ∈ V ( T ) \ { u } (cid:9) . By assigning one new color to eachpath in R \ R (cid:48) , it follows from (20) and the assumption b < a/ w ( T ) ≤ w ( T (cid:48) ) + b + b + · · · + b d < π ( T ) −
32 ( b + b + · · · + b d ) −
32 + ( b + b + · · · + b d )= 32 π ( T ) + b −
12 (1 + b + b + · · · + b d ) − < π ( T ) . When k = 2, R \ R (cid:48) is divided into the following classes. • P := (cid:8) P x,y : x = u , y ∈ V ( B (cid:48) ) ∪ { r } or x = u , y ∈ V ( B (cid:48) ) ∪ { r } (cid:9) . • P := { P x,y : x = u , y ∈ V ( B (cid:48) ) or x = u , y ∈ V ( B (cid:48) ) or x = u , y = u (cid:9) . • P i := { P x,y : x = u , y ∈ V ( B i ) or x = u , y ∈ V ( B i +1 ) } , for i = 3 , , . . . , d − • P d := { P x,y : x = u , y ∈ V ( B d ) or x = u , y ∈ V ( B ) } .One can check that χ ( Q ( P )) = b , χ ( Q ( P )) = |P | = 2 b − χ ( Q ( P d )) = b , and χ ( Q ( P i )) = b i for 3 ≤ i ≤ d −
1. It follows from (20) that w ( T ) ≤ w ( T (cid:48) ) + d (cid:88) i =1 χ ( Q ( P )) ≤ w ( T (cid:48) ) + 3 b + 2 b + b + b + · · · + b d − + 1 < π ( T ) − b −
32 ( b + b + · · · + b d ) −
32 + 3 b + 2 b + b + b + · · · + b d − + 1= 32 π ( T ) + 12 ( b − b − b − · · · − b d − − b d ) − < π ( T ) , (21)where (21) is due to b d − + b d > b and d ≥ k ≥ k is odd, R \ R (cid:48) is divided into the following classes. • P ( i,j ) := (cid:8) P u i ,y : y ∈ V ( B (cid:48) j ) (cid:9) , for 1 ≤ i, j ≤ k . • P := (cid:8) P u i ,r : 1 ≤ i ≤ k (cid:9) . • P i := (cid:8) P u i ,y : y ∈ V ( B k +1 ) (cid:9) , for 1 ≤ i ≤ k . • P ∞ := (cid:8) P u i ,u j : 1 ≤ i (cid:54) = j ≤ k (cid:9) . • (cid:98) P ( i,j ) := (cid:8) P u i ,y : y ∈ V ( B j ) (cid:9) , for 1 ≤ i ≤ k and k + 2 ≤ j ≤ d .Note that P ( i,i ) refers to the collection of paths connecting u i and vertices in B (cid:48) i . We remarkhere that |P | = k , |P ∞ | = (cid:0) k (cid:1) , and |P ( i,j ) | = b − |P i | = b k +1 ≤ b − | (cid:98) P ( i,j ) | = b j forall suitable i and j .Recall that the total-chromatic number of K k is k when k is odd. Let K k be a completegraph of k vertices labelled 1 , , . . . , k , and let f : V ( K k ) ∪ E ( K k ) → { , , . . . , k } be aproper k -total-coloring of K k . For t = 1 , , . . . , k denote by C t the collection of vertices andedges who receive color t under f . Notice that each C t contains exactly one vertex and k − edges. For the sake of argument, we assume vertex t ∈ C t . For any t , construct two HE UNDIRECTED OPTICAL INDICES OF TREES 13 sets O t and O rt as follows: For each edge { i, j } , i < j , put ( i, j ) and ( j, i ) into O t and O rt ,respectively.Pick two paths, one in P ( i,j ) and another in P ( i (cid:48) ,j (cid:48) ) , they can receive the same color if { i, j }∩{ i (cid:48) , j (cid:48) } = ∅ . For any t , since t / ∈ { i, j } and { i, j }∩{ i (cid:48) , j (cid:48) } = ∅ for any ( i, j ) , ( i (cid:48) , j (cid:48) ) ∈ O t ,we have χ Q P ( t,t ) ∪ (cid:91) ( i,j ) ∈O t P ( i,j ) ≤ max (cid:8) |P ( t,t ) | , |P ( i,j ) | (cid:9) ( i,j ) ∈O t = b − . Since paths in P t have no common edges with paths in P ( j,i ) , for any ( j, i ) ∈ O rt , by thesimilar argument we have χ Q P t ∪ (cid:91) ( j,i ) ∈O rt P ( j,i ) ≤ max (cid:8) |P t | , |P ( j,i ) | (cid:9) ( j,i ) ∈O rt = b − . By going through t from 1 up to k , it derives χ Q (cid:91) ≤ t ≤ k P t ∪ (cid:91) ≤ i,j ≤ k P ( i,j ) ≤ k ( b − . (22)The paths in P ∪ P ∞ can be dealt with in the same way. Any two paths in (cid:8) P t,r (cid:9) ∪ (cid:8) P u i ,u j : { i, j } ∈ C t (cid:9) have no common edges. This implies that χ ( Q ( P ∪ P ∞ )) = χ (cid:0) Q (cid:0)(cid:8) P t,r (cid:9) ∪ (cid:8) P u i ,u j : { i, j } ∈ C t (cid:9)(cid:1)(cid:1) ≤ k. (23)It remains to consider paths in (cid:98) P ( i,j ) , for 1 ≤ i ≤ k and k + 2 ≤ j ≤ d . Notice thatany two paths, one in (cid:98) P ( i,j ) and another in (cid:98) P ( i (cid:48) ,j (cid:48) ) , have no common edges if and only if i (cid:54) = i (cid:48) and j (cid:54) = j (cid:48) . We only consider the case when k ≤ d − k −
1, since the other case (i.e., k > d − k −
1) can be dealt with in the same way. The classes of paths can be arranged inthe following fashion. i S i (cid:98) P (1 ,k +2) , (cid:98) P (2 ,k +3) , (cid:98) P (3 ,k +4) , . . . , (cid:98) P ( k, k +1) .2 (cid:98) P (1 ,k +3) , (cid:98) P (2 ,k +4) , (cid:98) P (3 ,k +5) , . . . , (cid:98) P ( k, k +2) .... ... ... ... . . . ... d − k − (cid:98) P (1 ,d ) , (cid:98) P (2 ,k +2) , (cid:98) P (3 ,k +3) , . . . , (cid:98) P ( k, k ) .In general, set S t collects the classes of paths (cid:98) P (1 ,t + k +1) , (cid:98) P (2 ,t + k +2) , (cid:98) P (3 ,t + k +3) , . . . , (cid:98) P ( k,t +2 k ) ,where the addition is taken modulo d + 1 and plus k + 2. The chromatic number of theconflict graph induced by paths in S t is determined by the sizes of the classes (cid:98) P ( i,j ) therein;more precisely, χ ( Q ( S t )) ≤ max (cid:110) | (cid:98) P (1 ,t + k +1) | , | (cid:98) P (2 ,t + k +2) | , | (cid:98) P (3 ,t + k +3) | , . . . , | (cid:98) P ( k,t +2 k ) | (cid:111) = (cid:40) b k +2 , if t = 1 or d − k ≤ t ≤ d − k − b t + k +1 , otherwise . Thus, we have χ Q (cid:93) ≤ i ≤ k,k +2 ≤ j ≤ d (cid:98) P ( i,j ) ≤ d − k − (cid:88) t =1 χ ( Q ( S t ))= kb k +2 + b k +3 + b k +4 + · · · + b d − k +1 . (24)Combining (22)–(24), we obtain χ ( Q ( R \ R (cid:48) )) ≤ k ( b −
1) + k + kb k +2 + b k +3 + b k +4 + · · · + b d − k +1 , and further (20) yields w ( T ) ≤ w ( T (cid:48) ) + χ ( Q ( R \ R (cid:48) )) < π ( T ) − k − b − (cid:0) b k +1 + b k +2 + · · · + b d (cid:1) + 32 ( k − k ( b −
1) + k + kb k +2 + b k +3 + b k +4 + · · · + b d − k +1 = 32 π ( T ) − ( k − b − (cid:0) b k +1 (cid:1) + (cid:0) k − (cid:1) b k +2 + 12 ( k − − (cid:0) b k +3 + b k +4 + · · · + b d − k +1 (cid:1) − (cid:0) b d − k +2 + b d − k +3 + · · · + b d (cid:1) . (25)Since b k +2 ≤ b k +1 ≤ b − k ≥
3, one has − ( k − b − (cid:0) b k +1 (cid:1) + (cid:0) k − (cid:1) b k +2 + 12 ( k − ≤ − ( k −
3) + 12 ( k − < . (26)Therefore, the result follows by plugging (26) into (25).When k ≥ k is even, the argument is similar to the odd case with a slight modifi-cation. Let T (cid:48)(cid:48) be a tree obtained from T (cid:48) by removing an extra leave u k +1 from B k +1 , andlet B (cid:48) k +1 = B k +1 − { u k +1 } . By the same argument in (18)–(20), we have w ( T (cid:48)(cid:48) ) < π ( T ) −
32 (2 k − b − (cid:0) b k +1 + b k +2 + · · · + b d (cid:1) + 32 ( k − . (27)Let R (cid:48)(cid:48) be the all-to-all routing of T (cid:48)(cid:48) . R \ R (cid:48)(cid:48) can be divided into the following classes. • P ( i,j ) := (cid:8) P u i ,y : y ∈ V ( B (cid:48) j ) (cid:9) , for 1 ≤ i, j ≤ k + 1. • P := (cid:8) P u i ,r : 1 ≤ i ≤ k + 1 (cid:9) . • P i := (cid:8) P u i ,y : y ∈ V ( B (cid:48) k +2 ) (cid:9) , for 1 ≤ i ≤ k + 1. • P ∞ := (cid:8) P u i ,u j : 1 ≤ i (cid:54) = j ≤ k + 1 (cid:9) . • (cid:98) P ( i,j ) := (cid:8) P u i ,y : y ∈ V ( B j ) (cid:9) , for 1 ≤ i ≤ k + 1 and k + 3 ≤ j ≤ d .Observe that k + 1 is even. Again, by the same argument as proposed in (22)–(24), we have χ ( Q ( R \ R (cid:48)(cid:48) )) ≤ k + 1)( b −
1) + ( k + 1) + ( k + 1) b k +3 + b k +4 + b k +5 + · · · + b d − k . (28) HE UNDIRECTED OPTICAL INDICES OF TREES 15
Combining (27) and (28) yields w ( T ) ≤ w ( T (cid:48)(cid:48) ) + χ ( Q ( R \ R (cid:48)(cid:48) )) < π ( T ) − (cid:0) k − (cid:1) b − (cid:0) b k +1 + b k +2 (cid:1) + (cid:0) k − (cid:1) b k +3 + 12 ( k − − (cid:0) b k +4 + b k +5 + · · · + b d − k (cid:1) − (cid:0) b d − k +1 + b d − k +2 + · · · + b d (cid:1) . (29)It suffices to claim that − (cid:0) k − (cid:1) b − (cid:0) b k +1 + b k +2 (cid:1) + (cid:0) k − (cid:1) b k +3 + 12 ( k − < . (30)If b k +3 = 0, the left-and-side of (30) can be simplified as − (cid:0) ( k −
4) + 12 (cid:1) b + 12 (cid:0) ( k − − (cid:1) − (cid:0) b k +1 + b k +2 (cid:1) = −
12 ( k − b − −
12 ( b + 1) − (cid:0) b k +1 + b k +2 (cid:1) , which is less than 0 due to k ≥
4. If b k +3 >
0, by plugging b k +3 = b − (cid:15) , for some (cid:15) ≥ b k +1 ≥ b k +2 ≥ b k +3 = b − (cid:15) that − (cid:0) k − (cid:1) b − (cid:0) b k +1 + b k +2 (cid:1) + (cid:0) k − (cid:1) b k +3 + 12 ( k − ≤ − k(cid:15) + 12 k + 72 (cid:15) −
52 = − ( (cid:15) − (cid:0) k − (cid:1) −
12 ( k − < , as desired. (cid:3) Concluding Remarks
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HE UNDIRECTED OPTICAL INDICES OF TREES 17
School of Mathematical Sciences, Xiamen University, Xiamen 361005, China
E-mail address , Y.-H. Lo: [email protected]
Department of Applied Mathematics, National Chiao Tung University, Hsinchu 300, Taiwan,ROC
E-mail address , H.-L. Fu: [email protected]
School of Electronic and Optical Engineering, Nanjing University of Science and Tech-nology, Nanjing, ChinaNational Mobile Communications Research Laboratory, Southeast University, Nanjing,China
E-mail address , Y. Zhang: [email protected]
Department of Information Engineering, the Chinese University of Hong Kong, Shatin,Hong Kong
E-mail address , W. S. Wong:, W. S. Wong: