SKEW CHROMATIC INDEX of COMB, LADDER, and MOBIUS LADDER GRAPHS Joice Punitha M.1, S

International Journal of Pure and Applied Mathematics Volume 101 No. 6 2015, 1003-1011 ISSN: 1311-8080 (printed version); ISSN: 1314-3395 (on-line version) url: http://www.ijpam.eu AP ijpam.eu

SKEW CHROMATIC INDEX OF COMB, LADDER, AND MOBIUS LADDER GRAPHS Joice Punitha M.1, S. Rajakumari2 1Department of Mathematics L.N. Government College (Autonomous) Ponneri, 601 204, Tamilnadu, INDIA 2Department of Mathematics R.M. D Engineering College Kavaraipettai, 601 206, Tamilnadu, INDIA

Abstract: A skew edge coloring of a graph G is deﬁned to be a set of two edge colorings such that no two edges are assigned the same unordered pair of colors. The skew chromatic index s(G) is the minimum number of colors required for a skew edge coloring of G. In this paper, an algorithm is determined for skew edge coloring of comb, ladder and Mobius ladder graphs. Also the skew chromatic index of these graphs is solved in polynomial time.

AMS Subject Classiﬁcation: 05C15 Key Words: skew edge coloring; skew chromatic index; comb graph; ladder graph; Mobius ladder graph

1. Introduction

Let G = (V,E) be a simple, connected undirected graph with vertex set V and edge set E. An edge coloring of a graph G is an assignment of colors to the edges of G so that no two adjacent edges are assigned the same color. The minimum number of colors required for an edge coloring of G is the edge chromatic number denoted by χ′(G). By Vizing’s theorem, ∆(G) ≤ χ′(G) ≤ ∆(G) + 1, where ∆(G) is the maximum degree of vertices in G. Edge coloring problems are well studied in both computer science and mathematics [5] and [8]. In cellular c 2015 Academic Publications, Ltd. Received: March 12, 2015 url: www.acadpubl.eu 1004 J. Punitha M., S. Rajakumari communication, frequency reusing is done by modeling it as an edge coloring problem to avoid co-channel interference [11]. In this paper, we consider skew edge coloring problems which are inspired from the study of skew Room squares by R. A. Brualdi [4]. The concept of skew chromatic index was introduced by Marsha F. Foregger and better upper bounds for s(G) was discussed when G is cyclic, cubic or bipartite [6]. A skew edge coloring of G is an assignment of an ordered pair of colors (ai, bi) to each edge ei of G such that:

(i) the ai’s form an edge coloring of G,

(ii) the bi’s form an edge coloring of G, and

(iii)the pairs {ai, bi} are all distinct. The two edge colorings are referred to as component colorings of the skew edge coloring. The skew chromatic index s(G) is the minimum number of colors required for a skew edge coloring of G. In this paper, we have found an algorithm for skew edge coloring of comb, ladder and Mobius ladder graphs and the skew chromatic index of these graphs are solved in polynomial time. All the notations and deﬁnitions used in this paper are as in [3].

2. Lower Bound on s(G)

Skew chromatic index, s(G) is the minimum number of colors used in two edge colorings of G such that no two edges are assigned the same unordered pair of colors. Each component coloring of a skew edge coloring is itself an edge coloring. Therefore we have s(G) ≥ χ′(G). Since ∆(G) ≤ χ′(G) ≤ ∆(G) + 1, k + 1 we have s(G) ≥ ∆(G). If k colors are used then there are unordered 2 pairs of colors and this number must be at least as large as the number of edges k + 1 in G. Let k(m) denote the smallest integer k satisfying ≥ m where 2 m denotes the number of edges in G. Thus the best lower bound for s(G) is s(G) ≥ max{∆(G), k(| E(G) |)} [6].

3. Comb Graph

Deﬁnition 1. A graph G = (V,E) is called a comb graph [1] if (i) it is a tree (ii) all vertices are of degree at most three (iii) all the vertices of degree SKEW CHROMATIC INDEX OF COMB, LADDER... 1005 three lie on a single simple path.

Let us consider a comb graph with 2n vertices and m = 2n − 1 edges. For convenience, the edges are labeled as e1, e2, . . . , ek, ek+1, . . . , em in a speciﬁc manner as shown in Figure 1.

e2 e4 e6 ek ek+2 em-1 ...... e1 e3 e5 ek-1 ek+1 em-2 em

Figure 1: Comb graph with 2n vertices and m edges

Algorithm 2. Algorithm for skew edge coloring of comb graphs for n ≥ 4. Input: A comb graph with 2n vertices and m = 2n − 1 edges, n ≥ 4. k + 1 Step 1: Find the smallest positive integer k such that ≥ m. 2 Step 2: Let {1, 2, 3, . . . , k} be the set of colors available to color the edges k + 1 with unordered pairs of colors of the form {ai, bi} where ai’s form the 2 first component coloring and bi’s form the second component coloring. Step 3: Skew edge coloring can be done as two component colorings. th First component coloring: Starting with edge e1, i edge in every set of k edges of comb graph is colored using the ith color. When the number of edges in the last set is less than k, its ith edge is colored using ith color till all the edges of the graph are colored. This forms an edge coloring of the first component. See Figure 2. Second component coloring: Here, the 1st edge in j consecutive sets of k edges is colored with the 1st, 2nd, 3rd, ··· , jth color respectively. The remaining edges in each set are colored using the remaining colors continued from the corresponding next color from the set {1, 2, 3, . . . , k} until all the colors are exhausted. When the number of edges in the (j + 1)th set is less than k, then its first edge is colored with (j +1)th color and the subsequent edges are colored using the subsequent colors from the set {1, 2, 3, . . . , k} taken in order. This forms an edge coloring of the second component. See Figure 3. Step 4: Form the two component colorings (ai, bi) made of the first component coloring and the second component coloring of the ith edge. See Figure 4. Output: Skew edge coloring of the comb graph. 1006 J. Punitha M., S. Rajakumari

2 4 6 2 4 6 2 4 6 2

1 3 5 1 3 5 1 3 5 1 3

Figure 2: First component coloring of comb graph with 2n = 22, m = 21

2 4 6 3 5 1 4 6 2 5

1 3 5 2 4 6 3 5 1 4 6

Figure 3: Second component coloring of comb graph with 2n = 22, m = 21

2, 2 4, 4 6, 6 2, 3 4, 5 6, 1 2, 4 4, 6 6, 2 2, 5

1, 1 3,3 5, 5 1, 2 3, 4 5, 6 1, 3 3,5 5, 1 1, 4 3, 6

Figure 4: Skew edge coloring of comb graph with 2n = 22, m = 21 SKEW CHROMATIC INDEX OF COMB, LADDER... 1007

k + 1 Proof. Let e , e , e , . . . , em be the edges of G. Fix k such that ≥ m 1 2 3 2 k + 1 so that unordered pairs are available for skew edge coloring. The 2 coloring of edges is done in a specific order as in Figure 1. First k edges e1, e2, e3, . . . , ek are assigned the colors in such a way that all ordered pairs of the form (i, i), i = 1, 2, 3, . . . , k are used. In the second set of k edges, the first k − 1 edges are assigned the colors of the form (i, i + 1), i = 1, 2, 3, . . . , k − 1. As only k colors are considered, the ordered pair (k, k + 1) corresponding to (i = k) is not permissible. The kth edge is assigned the color (k, 1) which can be written as ((k − 1) + i, i), i = 1. In the next set of k edges, the first k − 2 edges are assigned colors of the form (i, i + 2), i = 1, 2, 3, . . . , k − 2. As the ordered pairs (k − 1, k + 1) and (k, k + 2) corresponding to (i = k − 1, k) are not permissible, the (k − 1)th edge and the kth edge are assigned the colors ((k − 2) + i, i), i = 1, 2 respectively. This process is continued till all the edges are colored and the above method of edge coloring assigns only distinct ordered pairs from the available k colors. See Figure 4.

Theorem 3. Let G be a comb graph with 2n vertices and m = 2n − 1 1+√1+8m edges, n ≥ 4. Then s(G) = k = − 2 . l m

Proof. As there are m edges, at least m ordered pairs of colors are required k + 1 for skew edge coloring of G. If k colors are used, then there will be 2 k + 1 pairs of colors. This must be at least as large as the number of edges 2 k + 1 k k in G. Therefore ﬁx k in such a way that ≥ m. i.e. ( +1) ≥ m . 2 2 It follows that k2 + k ≥ 2m. i.e.k2 + k − 2m ≥ 0. Thus solving for k and 1+√1+8m taking the positive root, we obtain k = − 2 ≥ 0, and its greatest integer 1+√1+8m k = − 2 will be the minimum number of colors used in skew edge l m 1+√1+8m coloring. Therefore s(G) = k = − 2 . l m Thus, based on the lower bound for s(G), we obtain an optimal solution for skew chromatic index of comb graphs for n ≥ 4. 1008 J. Punitha M., S. Rajakumari

4. Ladder Graph

Deﬁnition 4. A ladder graph Ln [2] is a planar undirected graph with 2n vertices and m = 3n − 2 edges where ∆(Ln) = 3 and δ(Ln) = 2.

For convenience, the edges are labeled as e1, e2, e3, . . . , ek, ek+1, . . . , em in a speciﬁc manner as shown in Figure 5.

e2 e5 e8 ek ek+3 em-2 ...... e1 e3 e6 ek+1 em-1

e e e e4 e7 e10 13 k-1 k+2 em

Figure 5: Generalized ladder graph

Algorithm 5. Algorithm for skew edge coloring of ladder graphs, n ≥ 6. Input: A ladder graph Ln with 2n vertices and m = 3n − 2 edges, n ≥ 6. Two component colorings of a ladder graph can be obtained in a manner similar to that of skew edge coloring of comb graph by following the steps 1 − 4 as in algorithm 2. See Figure 6. Output: Skew edge coloring of the ladder graph.

2, 2 5, 5 1, 2 4, 5 7, 1 3, 5 6, 1 2, 5 5, 1

1, 1 3,3 6, 6 2, 3 5, 6 1, 3 4, 6 7,2 3, 6 6, 2

4, 4 7, 7 3, 4 6, 7 2, 4 5, 7 1, 4 4, 7 7, 3

Figure 6: Skew edge coloring of ladder graph with 2n = 20, m = 28 and s(G) = 7

Theorem 6. Let Ln be a ladder graph with 2n vertices and m = 3n − 2 1+√24n 15 edges, n ≥ 6. Then s(G) = k = − 2 − . l m Thus, based on the lower bound for s(G), we prove that the skew chromatic index for ladder graph Ln is tight for n ≥ 6. SKEW CHROMATIC INDEX OF COMB, LADDER... 1009

5. Mobius Ladder Graph

A Mobius ladder graph Mn is a simple cubic graph on 2n vertices and 3n edges [10]. Mobius ladder graph was introduced by Richard Guy and Frank Harary in 1966 [7].

Deﬁnition 7. A Mobius ladder graph Mn [9] is a graph obtained from the ladder Pn × P2 by joining the opposite end points of the two copies of Pn. For convenience, the edges are labelled as e1, e2, e3, . . . , ek, ek+1, . . . , em 1, em in a − speciﬁc manner as shown in Figure 7.

e2 e5 e8 ek ek+3 em-4 e ...m ... e1 e3 e6 ek+1 em-3

e m-1 e e e4 e7 k-1 k+2 em-2

Figure 7: Generalized Mobius ladder graph

Algorithm 8. Algorithm for skew edge coloring of Mobius ladder graphs, n ≥ 6. Input: A Mobius ladder graph Mn with 2n vertices and m = 3n edges, n ≥ 6. Step 1: Consider the subgraph of Mobius ladder graph Mn without the diagonal edges which is a ladder graph Ln with 2n vertices and 3n − 2 edges. By proceeding as in algorithm 5, obtain the two component colorings (ai, bi) of the subgraph Ln for each of its edges ei, i = 1, 2, . . . , m − 2. Step 2: Obtain the pairs of colors to color the edges em 1 and em based − on the coloring components of the edge em 2. − Case 1: When both the coloring components of em 2 namely (am 2, bm 2) − − − are less than k−1, then form the pairs (am 2 +1, bm 2 +1), (am 2 +2, bm 2 +2). The two pairs thus formed are distinct with− each− of the colori−ng components− being less than or equal to k. Case 2: When atleast one of the coloring components of the edge em 2 − is either k or k − 1, then consider the coloring of the edges em 2, em 3, em 4 − − − viz. (am 2, bm 2), (am 3, bm 3), (am 4, bm 4) and form the pairs (bm 2, am 3), − − − − − − − − (bm 3, am 4) and if possible, obtain the ordered pairs (bm 2+1, am 3+1), (bm 3+ − − − − − 1, am 4 + 1) provided each of the coloring components is less than or equal to − 1010 J. Punitha M., S. Rajakumari k. Out of these four pairs, two or three are distinct. Step 3: Assign colors to the edges em 1 and em based on the components of the pairs obtained in step 2. − Case 1: If one of the coloring components is 1, do not use that pair for coloring the edges. Case 2: If the coloring components are independent of the colors 2 or 4, then assign those pairs to the edges em 1 and em taken in order. − Case 3: If one of the coloring components is 2, then assign it to the edge em 1 and the next pair to the edge em. − Case 4: If one of the coloring components is 4, then assign it to the edge em and the next pair to the edge em 1. Output: Skew edge coloring of the− Mobius ladder graph. See Figure 8. 2, 2 5, 5 8, 8 3, 4 6, 7 1, 3 4, 6 7, 1 2 5 7, 3 1, 1 3,3 6, 6 1, 2 4, 5 7, 8 2, 4 5,7 8, 2 3, 6 6, 2 4, 4 7, 7 2, 3 5, 6 8, 1 3, 5 6, 8 1, 4 4, 7

Figure 8: Skew edge coloring of Mobius ladder graph with 2n = 20, m = 30 and s(G) = 8

Theorem 9. Let Mn be a Mobius ladder graph with 2n vertices and 1+√1+24n m = 3n edges, n ≥ 6. Then s(G) = − 2 . l m Thus, based on the lower bound for s(G), we obtain an optimal solution for skew chromatic index of Mobius ladder graphs for n ≥ 6.

6. Conclusion

In this paper, we have designed algorithms for skew edge coloring of comb, ladder and Mobius ladder graphs and obtained an optimal solution for s(G). It would be interesting to identify the skew chromatic index for various networks.

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