Department of Pure Mathematics, Ferdowsi University of Mashhad (in cooperation with the Center of Excellence in Analysis on Algebraic Structures and Tusi Mathematical Research Group)Khayyam Journal of Mathematics2423-47886220200701Strong rainbow coloring of unicyclic graphs20621610981810.22034/kjm.2020.109818ENAminRostamiDepartment of Pure Mathematics, Ferdowsi University
of Mashhad, P.O. Box 1159, Mashhad 91775, IranMadjidMirzavaziriDepartment of Pure Mathematics, Ferdowsi University
of Mashhad, P.O. Box 1159, Mashhad 91775, IranFreydoonRahbarniaDepartment of Applied Mathematics, Ferdowsi
University of Mashhad, P.O. Box
1159, Mashhad 91775, Iran.Journal Article20190619A path in an edge-colored graph is called a \textit{rainbow path}, if no two edges of the path are colored the same. An edge-colored graph $G$, is \textit{rainbow-connected} if any two vertices are connected by a rainbow path. A rainbow-connected graph is called strongly rainbow connected if for every two distinct vertices $u$ and $v$ of $V(G)$, there exists a rainbow path $P$ from $u$ to $v$ that in the length of $P$ is equal to $d(u,v)$. The notations {\rm rc}$(G)$ and {\rm src}$(G)$ are the smallest number of colors that are needed in order to make $G$ rainbow connected and strongly rainbow connected, respectively. In this paper, we find the exact value of {\rm rc}$(G)$, where $G$ is a unicyclic graph. Moreover, we determine the upper and lower bounds for {\rm src}$(G)$, where $G$ is a unicyclic graph, and we show that these bounds are sharp.http://www.kjm-math.org/article_109818_301fa61eefbf5cce342037ce7790283b.pdf