The rainbow connection of a graph is (at most) reciprocal to its minimum degree
نویسندگان
چکیده
An edge-colored graph G is rainbow edge-connected if any two vertices are connected by a path whose edges have distinct colors. The rainbow connection of a connected graph G, denoted by rc(G), is the smallest number of colors that are needed in order to make G rainbow edgeconnected. We prove that if G has n vertices and minimum degree δ then rc(G) < 20n/δ. This solves open problems from [5] and [3]. A vertex-colored graph G is rainbow vertex-connected if any two vertices are connected by a path whose internal vertices have distinct colors. The rainbow vertex-connection of a connected graph G, denoted by rvc(G), is the smallest number of colors that are needed in order to make G rainbow vertex-connected. One cannot upper-bound one of these parameters in terms of the other. Nevertheless, we prove that if G has n vertices and minimum degree δ then rvc(G) < 11n/δ. We note that the proof in this case is different from the proof for the edgecolored case, and we cannot deduce one from the other.
منابع مشابه
Rainbow connection and minimum degree
An edge-coloured graph G is rainbow connected if any two vertices are connected by a path whose edges have distinct colours. The rainbow connection number of a connected graph G, denoted rc(G), is the smallest number of colours that are needed in order to make G rainbow connected. Krivelevich and Yuster have shown that a connected graph G with n vertices has rc(G) < 20n δ(G) [M. Krivelevich and...
متن کاملDiameter Two Graphs of Minimum Order with Given Degree Set
The degree set of a graph is the set of its degrees. Kapoor et al. [Degree sets for graphs, Fund. Math. 95 (1977) 189-194] proved that for every set of positive integers, there exists a graph of diameter at most two and radius one with that degree set. Furthermore, the minimum order of such a graph is determined. A graph is 2-self- centered if its radius and diameter are two. In this paper for ...
متن کاملProduct version of reciprocal degree distance of composite graphs
A {it topological index} of a graph is a real number related to the graph; it does not depend on labeling or pictorial representation of a graph. In this paper, we present the upper bounds for the product version of reciprocal degree distance of the tensor product, join and strong product of two graphs in terms of other graph invariants including the Harary index and Zagreb indices.
متن کاملRainbow Connection in Graphs with Minimum Degree Three
An edge-coloured graph G is rainbow connected if any two vertices are connected by a path whose edges have distinct colours. This concept of rainbow connection in graphs was recently introduced by Chartrand et al.. The rainbow connection number of a connected graph G, denoted rc(G), is the smallest number of colours that are needed in order to make G rainbow connected. The computation of rc(G) ...
متن کاملOn Rainbow Connection Number and Connectivity
Rainbow connection number, rc(G), of a connected graph G is the minimum number of colours needed to colour its edges, so that every pair of vertices is connected by at least one path in which no two edges are coloured the same. In this paper we investigate the relationship of rainbow connection number with vertex and edge connectivity. It is already known that for a connected graph with minimum...
متن کاملذخیره در منابع من
با ذخیره ی این منبع در منابع من، دسترسی به آن را برای استفاده های بعدی آسان تر کنید
برای دانلود متن کامل این مقاله و بیش از 32 میلیون مقاله دیگر ابتدا ثبت نام کنید
ثبت ناماگر عضو سایت هستید لطفا وارد حساب کاربری خود شوید
ورودعنوان ژورنال:
- Journal of Graph Theory
دوره 63 شماره
صفحات -
تاریخ انتشار 2010