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New Algorithm for Calculating Chromatic Index of Graphs and Its Applications F Salama Journal of the Egyptian Mathematical Society (2019) 27:18 Journal of the Egyptian https://doi.org/10.1186/s42787-019-0018-9 Mathematical Society ORIGINALRESEARCH Open Access New algorithm for calculating chromatic index of graphs and its applications F. Salama1,2 Correspondence: fatma.salama@ science.tanta.edu.eg Abstract 1Department of Mathematics, Faculty of Science, Tanta University, The problem of edge coloring is one of the fundamental problems in the graph Tanta, Egypt theory. Graph coloring problem arises in several ways like to allocate scheduling and 2Department of Mathematics, assignments. To follow this line of investigation, we design a new algorithm called Faculty of Science, Taibah “ ” University, Madinah, Kingdom of RF algorithm to color the edges of a graph. In addition, we reinstate some classical Saudi Arabia results by applying the RF algorithm. Keywords: Graph, Edge coloring, Chromatic index, Incidence matrix 2010 Mathematics subject classification: 05C15, 05C99, 05C90, 05C30. Introduction The problem of edge coloring appeared with the four-color problem. In 1880, Tait wrote the first paper dealing with the problem of edge coloring. Tait proved that only three colors are used to color the edges of every 3-connected planar graph. An s-edge coloring, s is a positive integer, is a way to color edges with s colors. The chromatic index χ ΄ (x)isthe minimum number of different colors needed to color edges such that any two adjacent edges are colored by different colors (for more details, see [1, 3–5, 7–9, 11–14]). K nig χ ΄ Δ 3ΔðxÞ has proved, in 1916, that (x)= (x) for every bipartite graph. 2 is the most number of colors that any graph can be edge-colored with, this was given by Shannon. Moreover, Viz- ing obtained for any simple graph Δ(x)+1≥ χ ΄ (x). Omai et al. [10]determinedtheAVD- chromatic index for the powers of paths. Further, Lehner [6] established a result state that if every non-trivial automorphism of a countable graph G with distinguishing index D ΄ (G) moves infinitely many edges, then D ΄ (G) ≤ 2. Grzesik and Khachatrian [2]provedthatk1, m, n is interval colorable iff gcd(m +1,n +1)=1. The above discussions motivate us to design a new algorithm to calculate the chromatic index of the graph. The main objective of our algorithm is to find a coloring that uses the smal- lest possible number of distinct colors. Some examples are given in support of our algorithm. The main results In this article, a new algorithm, RF coloring algorithm, will be designed to evaluate chro- matic index of loopness graph. RF coloring algorithm is introduced as follows: Consider a graph G of order n and size m. List its vertices as v1, v2, v3, … , vn and its edges as e1, e2, e3, … , em. © The Author(s). 2019 Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made. Salama Journal of the Egyptian Mathematical Society (2019) 27:18 Page 2 of 8 Step1. Write the incidence matrix of the graph G 2 3 a11 a12 … a1m 6 … 7 a21 a22 a2m : IGðÞ¼4 ⋮ ⋮ ⋱ ⋮ 5 an1 an2 … anm Step2. In this step, we constitute the RF coloring matrix from the incidence matrix as follows: (a) In the first row of the RF matrix, put 8 < 0if a11 ¼ 0 à a11 ¼ : 1if a11 ¼ 1 After that put 8 < 0if a12 ¼ 0 à ; a12 ¼ : 1ifa12 ¼ 1 a11 ¼ 0 2if a11 ¼ a12 ¼ 1 à ≤ In the same way, for entry a1 j, where 1 < j m, put 8 < 0ifa1 j ¼ 0 aà ¼ no 1 j : à ; à ; …; à max a11 a12 a1ðÞj−1 þ 1ifa1 j ¼ 1 k à (b) Any column has the entry a1k ¼ h put 8 < h if aik ¼ 1 à aik ¼ : 0ifaik ¼ 0 where 1 < i ≤ n and 1 < k ≤ m. (c) Now start from the second row put 8 à > a ¼ 0if a2l ¼ 0 <> no2nol nono à à à à à à à à à à à à  min a : a ≠0; n a ; a : a ≠0; a ≠0 if a2l ¼ 1; aSl ¼ 1; a : a ≠0 n a ; a : a ≠0≠a ≠o > no1 j 1 j 2 j sj 2 j sj no1 j 1 j no2 j Sj 2 j Sj > : à ; à ; à : à ≠ ; à ≠ ≠ à ; ; ; à : à ≠ à ; à : à ≠ ≠ à  max a1 j a2 j asj a1 j 0 a2 j 0 asj þ 1ifa2l ¼ 1 aSl ¼ 1 a1 j a1 j 0 n a2 j aSj a2 j 0 aSj ¼ o where 1 ≤ l ≤ m,1≤ j ≤ m and 1 ≤ s ≤ n. Salama Journal of the Egyptian Mathematical Society (2019) 27:18 Page 3 of 8 l à (d) Any column has the entry a2l ¼ f put 8 < f if ail ¼ 1 à ail ¼ : 0ifail ¼ 0 (e) Again repeat steps (c) and (d) to complete the RF coloring matrix 2 3 à à … à a11 a12 a1m 6 à à … à 7 6 a21 a22 a2m 7 RCIðÞ¼G 4 ⋮ ⋮ ⋱ ⋮ 5 à à … à an1 an2 anm Step3. The greatest number in the RF coloring matrix is the chromatic index of the G à Ã≠ graph and the value of entry aij is the color of the edge e j where aij 0. In the following, we give examples solved by the new algorithm: Example1. Let G be a graph shown below (Fig. 1). The chromatic index of the graph G will be calculated by using the RF coloring algo- rithm as follows: 2 3 2 3 Since a11 ¼ 1 2 3 11 10 Since a11 ¼ 1 1 a12 ¼ 1 12 à à 6 11 017 Put a11 ¼ 1 6 7 Put a12 ¼ 2 6 7 IGðÞ¼4 5 → 4 5 → 4 5 0 0 1 0 0 0 0 1 Fig. 1 The graph G Salama Journal of the Egyptian Mathematical Society (2019) 27:18 Page 4 of 8 2 3 2 3 Column 1 2 3 12 3 12 30 à ; 1230 Since a14 ¼ 0 Put a21 ¼ 1 6 7 Put aà ¼ 0 6 7 aà ¼ aà ¼ 0 6 7 Since a a a 1Put aà 34 5 →14 4 5 31 →41 4 1 5 11 ¼ 12 ¼ 13 ¼ 13 ¼ 0 → 0 Column 2 2 3 Column3 à ; 1230 à ; Put a22 ¼ 2 Put a33 ¼ 3 aà ¼ aà ¼ 0 6 7 aà ¼ aà ¼ 0 32 →42 4 12 5 23 →43 0 0 0 0 2 3 2ndrow 2 3 1230 Since a24 ¼ 1 1230 6 7 then aà ¼ maxfgn 1; 2; 3 fg1; 2 ¼ 3 6 7 4 12 0 5 24 → 4 12035 0 0 3 0 0 3 0 0 0 0 0 0 Column 4 2 3 à ; 12 30 Put a44 ¼ 3 aà ¼ 0 6 7 43→ 4 12 035 0 0 3 0 1 2 0 3 Then, the RF coloring matrix is given by 2 3 12 30 6 12 037 RCIðÞ¼G 4 5: 0 0 3 0 1 2 0 3 From the above matrix, we find the chromatic index χ ΄ (G) of the graph G is equal to 3. Example2. Given a graph G, shown in Fig. 2. Fig. 2 The quartic graph Salama Journal of the Egyptian Mathematical Society (2019) 27:18 Page 5 of 8 Applying the RF coloring algorithm step by step to evaluate the colors of edges and chromatic index as shown in the sequence of matrices below: Hence, the RF coloring matrix is given by 2 3 1 0 2 0 3 4 0 0 0 0 0 0 6 7 6 1 3 0 2 0 0 0 0 4 0 0 0 7 6 0 3 2 0 0 0 1 5 0 0 0 0 7 RCIðÞ¼G 6 7: 6 0 0 0 2 3 0 0 0 0 1 5 0 7 4 0 0 0 0 0 0 0 5 4 1 0 2 5 0 0 0 0 0 4 1 0 0 0 5 2 It is clear from the RF coloring matrix that the chromatic index of the graph G is 5, i.e., χ ΄ (G) = 5 and the color of e1, e7, e10 is 1, the color of e3, e4, e12 is 2, the color of e2, e5 is 3, the color of e6, e9 is 4, and the color of e8, e11 is 5. In the following section, we reprove some theorems by using RF coloring algorithm Theorem 1 Let Sn be a star graph of order n. Then, χ ΄ (Sn)=n − 1, where χ ΄ (Sn) is the chromatic index of Sn. Proof Let Sn be a star graph of order n as shown in Fig. 3. By applying the RF coloring algorithm, we found the RF coloring matrix is given by 2 3 123 … n−1 6 … 7 6 1 00 0 7 RCIðÞ¼Sn 6 0 20 … 0 7 4 ⋮ ⋮⋮ ⋱ ⋮ 5 0 00 … n−1 The greatest number in the RF coloring matrix is n − 1, then the chromatic index of − Sn equals n 1. 8 < 2 if n is even χ΄ Theorem 2 Let Cn be a cycle graph of order n. Then ðCnÞ¼: , 3 if n is odd where χ ΄ (Cn) is the chromatic index of Cn. Salama Journal of the Egyptian Mathematical Society (2019) 27:18 Page 6 of 8 Fig.
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