On proper (strong) rainbow connection of graphs

Abstract

A path in an edge-colored graph G is called a rainbow path if no two edges on the path have the same color. The graph G is called rainbow connected if between every pair of distinct vertices of G, there is a rainbow path. Recently, Johnson et al. considered this concept with the additional requirement that the coloring of G is proper. The proper rainbow connection number of G, denoted by prc(G), is the minimum number of colors needed to properly color the edges of G so that G is rainbow connected. Similarly, the proper strong rainbow connection number of G, denoted by psrc(G), is the minimum number of colors needed to properly color the edges of G such that for any two distinct vertices of G, there is a rainbow geodesic (shortest path) connecting them. In this paper, we characterize those graphs with proper rainbow connection numbers equal to the size or within 1 of the 2 H. Jiang, W. Li, X. Li and C. Magnant size. Moreover, we completely solve a question proposed by Johnson et al. by proving that if G = K p1 • • • K pn , where n ≥ 1, and p 1 ,. .. , p n > 1 are integers, then prc(G) = psrc(G) = χ ′ (G), where χ ′ (G) denotes the chromatic index of G. Finally, we investigate some sufficient conditions for a graph G to satisfy prc(G) = rc(G), and make some slightly positive progress by using a relation between rc(G) and the girth of the graph.

Proper rainbow connection number of graphs

“…The authors in [7] determined some graphs with large proper rainbow connection number. First of all, they characterized all graphs whose proper connection numbers equal their size.…”

“…First of all, they characterized all graphs whose proper connection numbers equal their size. Theorem 1.3 (Jiang et al [7]) Let G be a connected graph of size m. Then prc(G) = m if and only if G is a tree or K 3 .…”

“…Let H ′ and H ′′ be two graph classes as shown in Figure 1, where the order of H ′ ∈ H ′ is at least 4 and the order of H ′′ ∈ H ′′ is at least 5, respectively. Theorem 1.4 (Jiang et al [7])…”

“…Next, the proper rainbow connection numbers of special graphs were considered by Bau et al [1] and Jiang et al [7].…”

“…The cartesian product of two graphs G and H written G H is the graph with vertex set V (G) × V (H) specified by putting (u 1 , u 2 ) adjacent to (v 1 , v 2 ) if and only if (1) [1] and Jiang et al [7] determined proper connection numbers of cartesian products by the following results.…”