Rainbow connection numbers of Cartesian product of graphs

• Main Authors: Yu-Jung Liang, 梁育榮
• Other Authors: David Kuo
• Format: Others
• Published: 2012
• Online Access: http://ndltd.ncl.edu.tw/handle/4xzvg3

碩士 === 國立東華大學 === 應用數學系 === 100 === Given a connected graph G together with a coloring f from the edge set of G to a set of colors, where adjacent edges may be colored the same, a u-v path P in G is said to be a rainbow path if no two edges of P are colored the same. A u-v path P in G is said to be a rainbow u-v geodesic in G if P is a rainbow u-v path whose length equals to the distance of u and v. The graph G is rainbow-connected(resp., strongly rainbow-connected) if G contains a rainbow u-v path(resp,. rainbow u-v geodesic) for every two vertices u and v of G. In this case, the coloring f is called a rainbow coloring(resp,. strong rainbow coloring) of G. A rainbow coloring(resp., strong rainbow coloring) of G using k colors is a rainbow k-coloring(resp., strong rainbow k-coloring) of G. The minimum k for which there exists a rainbow k-coloring(resp., strong rainbow k-coloring) of G is called the rainbow connection number(resp., strong rainbow connection number) of G and is denoted by rc(G)(resp., src(G)). We study the rainbow connection numbers and the strong rainbow connection numbers of Cartesian product of graphs, where both of the two graphs are in F={G:G is a path, a cycle, or a complete graph}, or both of the two graphs are in T={T:T is a tree}, in this thesis. We show that if G is the Cartesian product of two graphs G₁ and G₂, in F, then diam(G)=rc(G)=src(G), except that both G₁ and G₂ are odd cycles. And we prove that if G is the Cartesian product of two trees T₁ and T₂, then rc(G)=diam(G), except that T₂ is the path P₂, and T₁ satisfies some special conditions, in which case the rainbow connection number of G equals diam(G)+1.