Rainbow Connection Number and Connected Dominating Sets

Rainbow connection number rc(G) of a connected graph G is the minimum number of colours needed to colour the edges of G, 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 show that for every connected graph G, with minimum degree at least 2, the rainbow connection number is upper bounded by {\gamma}_c(G) + 2, where {\gamma}_c(G) is the connected domination number of G. Bounds of the form diameter(G) \leq rc(G) \leq diameter(G) + c, 1 \leq c \leq 4, for many special graph classes follow as easy corollaries from this result. This includes interval graphs, AT-free graphs, circular arc graphs, threshold graphs, and chain graphs all with minimum degree at least 2 and connected. We also show that every bridge-less chordal graph G has rc(G) \leq 3.radius(G). In most of these cases, we also demonstrate the tightness of the bounds. An extension of this idea to two-step dominating sets is used to show that for every connected graph on n vertices with minimum degree {\delta}, the rainbow connection number is upper bounded by 3n/({\delta} + 1) + 3. This solves an open problem of Schiermeyer (2009), improving the previously best known bound of 20n/{\delta} by Krivelevich and Yuster (2010). Moreover, this bound is seen to be tight up to additive factors by a construction of Caro et al. (2008).Comment: 14 page

Color Code Techniques In Rainbow Connection

Let G be a graph with an edge k-coloring γ : E(G) → {1, …, k} (not necessarily proper). A path is called a rainbow path if all of its edges have different colors. The map γ is called a rainbow coloring if any two vertices can be connected by a rainbow path. The map γ is called a strong rainbow coloring if any two vertices can be connected by a rainbow geodesic. The smallest k for which there is a rainbow k-coloring (resp. strong rainbow k-coloring) on G is called the rainbow connection number (resp. strong rainbow connection number) of G, denoted rc(G) (resp. src(G)). In this paper we generalize the notion of “color codes” that was originally used by Chartrand et al. in their study of the rc and src of complete bipartite graphs, so that it now applies to any connected graph. Using color codes, we prove a new class of lower bounds depending on the existence of sets with common neighbours. Tight examples are discussed, involving the amalgamation of complete graphs, generalized wheel graphs, and a special class of sequential join of graphs

Algorithms and Bounds for Very Strong Rainbow Coloring

A well-studied coloring problem is to assign colors to the edges of a graph $G$ so that, for every pair of vertices, all edges of at least one shortest path between them receive different colors. The minimum number of colors necessary in such a coloring is the strong rainbow connection number (\src(G)) of the graph. When proving upper bounds on \src(G), it is natural to prove that a coloring exists where, for \emph{every} shortest path between every pair of vertices in the graph, all edges of the path receive different colors. Therefore, we introduce and formally define this more restricted edge coloring number, which we call \emph{very strong rainbow connection number} (\vsrc(G)). In this paper, we give upper bounds on \vsrc(G) for several graph classes, some of which are tight. These immediately imply new upper bounds on \src(G) for these classes, showing that the study of \vsrc(G) enables meaningful progress on bounding \src(G). Then we study the complexity of the problem to compute \vsrc(G), particularly for graphs of bounded treewidth, and show this is an interesting problem in its own right. We prove that \vsrc(G) can be computed in polynomial time on cactus graphs; in contrast, this question is still open for \src(G). We also observe that deciding whether \vsrc(G) = k is fixed-parameter tractable in $k$ and the treewidth of $G$. Finally, on general graphs, we prove that there is no polynomial-time algorithm to decide whether \vsrc(G) \leq 3 nor to approximate \vsrc(G) within a factor $n^{1-\varepsilon}$, unless P$=$NP

The strong rainbow vertex-connection of graphs

A vertex-colored graph $G$ is said to be rainbow vertex-connected if every two vertices of $G$ are connected by a path whose internal vertices have distinct colors, such a path is called a rainbow path. The rainbow vertex-connection number 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. If for every pair $u, v$ of distinct vertices, $G$ contains a rainbow $u-v$ geodesic, then $G$ is strong rainbow vertex-connected. The minimum number $k$ for which there exists a $k$-vertex-coloring of $G$ that results in a strongly rainbow vertex-connected graph is called the strong rainbow vertex-connection number of $G$, denoted by $srvc(G)$. Observe that $rvc(G)\leq srvc(G)$ for any nontrivial connected graph $G$. In this paper, sharp upper and lower bounds of $srvc(G)$ are given for a connected graph $G$ of order $n$, that is, $0\leq srvc(G)\leq n-2$. Graphs of order $n$ such that $srvc(G)= 1, 2, n-2$ are characterized, respectively. It is also shown that, for each pair $a, b$ of integers with $a\geq 5$ and $b\geq (7a-8)/5$, there exists a connected graph $G$ such that $rvc(G)=a$ and $srvc(G)=b$.Comment: 10 page

Rainbow Colorings in Graphs

In this thesis, we deal with rainbow colorings of graphs. We engage not with the rainbow connection number but with counting of rainbow colorings in graphs with k colors. We introduce the rainbow polynomial and prove some results for some special graph classes. Furthermore, we obtain bounds for the rainbow polynomial. In addition, we define some edge colorings related to the rainbow coloring, like the s-rainbow coloring and the 2-rainbow coloring. For this edge colorings, polynomials are defined and we prove some basic properties for this polynomials and present some formulas for the calculation in special graph classes. In addition, we consider in this thesis counting problems related to the rainbow coloring like rainbow pairs and rainbow dependent sets. We introduce polynomials for this counting problems and present some general properties and formulas for special graph classes