Computing Minimum Rainbow and Strong Rainbow Colorings of Block Graphs

A path in an edge-colored graph $G$ is rainbow if no two edges of it are colored the same. The graph $G$ is rainbow-connected if there is a rainbow path between every pair of vertices. If there is a rainbow shortest path between every pair of vertices, the graph $G$ is strongly rainbow-connected. The minimum number of colors needed to make $G$ rainbow-connected is known as the rainbow connection number of $G$, and is denoted by $\text{rc}(G)$. Similarly, the minimum number of colors needed to make $G$ strongly rainbow-connected is known as the strong rainbow connection number of $G$, and is denoted by $\text{src}(G)$. We prove that for every $k \geq 3$, deciding whether $\text{src}(G) \leq k$ is NP-complete for split graphs, which form a subclass of chordal graphs. Furthermore, there exists no polynomial-time algorithm for approximating the strong rainbow connection number of an $n$-vertex split graph with a factor of $n^{1/2-\epsilon}$ for any $\epsilon > 0$ unless P = NP. We then turn our attention to block graphs, which also form a subclass of chordal graphs. We determine the strong rainbow connection number of block graphs, and show it can be computed in linear time. Finally, we provide a polynomial-time characterization of bridgeless block graphs with rainbow connection number at most 4.Comment: 13 pages, 3 figure

RAINBOW CONNECTION AND STRONG RAINBOW CONNECTION NUMBERS OF

A rainbow path in an edge coloring of graph  is a path in which every two edges are assigned different colors. If a nontrivial connected graph  contains a rainbow path for every two vertices in , then  is rainbow-connected. The rainbow connection number  of  is the minimum integer  such that  is rainbow connnected in an edge-coloring with  colors. If a nontrivial connected  contains a rainbow path  geodesic for every two vertices in , then  is strongly rainbow-connected. The strong rainbow connection number  of  is the minimum integer  such that  is strongly rainbow-connected in an edge coloring with  colors. The rainbow connection and strong rainbow connection number of graph , , and  have been found. In this paper, the results will be generalized and the rainbow connection and strong rainbow connection number of graph  will be determined. Keywords: rainbow coloring, rainbow connection number, strong rainbow coloring, strong rainbow connection number, grap

The (Strong) Rainbow Connection Number of Join Of Ladder and Trivial Graph

Let G = (V,E) be a nontrivial, finite, and connected graph. A function c from E to {1,2,...,k},k ∈ N, can be considered as a rainbow k-coloring if every two vertices x and y in G has an x- y path. Therefore, no two path's edges receive the same color; this condition is called a “rainbow path”. The smallest positive integer k, designated by rc(G), is the G rainbow connection number. Thus, G has a rainbow k-coloring. Meanwhile, the c function is considered as a strong rainbow k-coloring within the condition for every two vertices x and y in G have an x - y rainbow path whose length is the distance between x and y. The smallest positive integer k, such as G, has a strong rainbow k-coloring; such a condition is called a strong rainbow connection number of G, denoted by src(G). In this research, the rainbow connection number and strong rainbow connection number are determined from the graph resulting from the join operation between the ladder graph and the trivial graph, denoted by rc(L_n∨K_1) and src(L_n∨K_1) respectively. So, rc (L_n∨K_1 )= src (L_n∨K_1 )=2,"for" 3≤n≤4 and rc (L_n∨K_1 )=3, while src(L_n∨K_1 )=⌈n/2⌉,"for" n≥5.

Penentuan Rainbow Connection Number dan Strong Rainbow Connection Number pada Graf berlian.

Misalkan G = (V,E) adalah suatu graf. Suatu pewarnaan c : E(G) → {1,2,...,k}, k ∈ N pada graf G adalah suatu pewarnaan sisi di G sedemikian sehingga setiap sisi bertetangga boleh berwarna sama. Misalkan u,v ∈ V (G) dan P adalah suatu lintasan dari u ke v. Suatu lintasan P dikatakan rainbow path jika tidak ter dapat dua sisi di P berwarna sama. Graf G disebut rainbow connected dengan pewarnaan c jika untuk setiap u,v ∈ V (G) terdapat rainbow path dari u ke v. Jika terdapat k warna di G maka c adalah rainbow k-coloring. Rainbow connec tion number dari graf terhubung dinotasikan dengan rc(G), didefenisikan sebagai banyaknya warna minimal yang diperlukan untuk membuat graf G yang bersi fat rainbow connected. Selanjutnya, pewarnaan c dikatakan pewarnaan-k strong rainbow, jika untuk setiap u dan v di V terdapat lintasan pelangi dengan pan jangnya sama dengan jarak u dan v. Dalam skripsi ini akan ditentukan rainbow connection number dan strong rainbow connection number pada graf berlian 2n titik dinotasikan dengan Brn adalah graf yang diperoleh dari graf tangga segitiga dengan 2n − 1 titik ditambahkan satu titik dan beberapa sisi tertentu. Dalam skripsi ini menentukan rc(Brn) dan src(Brn) untuk n ≥ 4 . Kata kunci : Rainbow connection number, strong rainbow connection number, graf berlian, graf path, Pewarnaan rainbow