Rainbow k -connection in Dense Graphs Shinya Fujita 1 , Henry Liu 2 - PowerPoint PPT Presentation

Rainbow k -connection in Dense Graphs Shinya Fujita 1 , Henry Liu ∗ 2 , Colton Magnant 3 1 Gunma National College of Technology, Japan 2 Universidade Nova de Lisboa, Portugal 3 Georgia Southern University, GA, USA EuroComb’11, Budapest, August/September 2011

Introduction Graphs with Fixed Connectivity Complete Bipartite and Multipartite Graphs Random Graphs Open Problems Introduction ◮ G is a finite, simple, k -connected graph ( k ∈ N ). Shinya Fujita, Henry Liu ∗ , Colton Magnant Rainbow k -connection in Dense Graphs

Introduction Graphs with Fixed Connectivity Complete Bipartite and Multipartite Graphs Random Graphs Open Problems Introduction ◮ G is a finite, simple, k -connected graph ( k ∈ N ). ◮ An edge-coloured path is rainbow if its edges have distinct colours. Shinya Fujita, Henry Liu ∗ , Colton Magnant Rainbow k -connection in Dense Graphs

Introduction Graphs with Fixed Connectivity Complete Bipartite and Multipartite Graphs Random Graphs Open Problems Introduction ◮ G is a finite, simple, k -connected graph ( k ∈ N ). ◮ An edge-coloured path is rainbow if its edges have distinct colours. ◮ An edge-colouring (not necessarily proper) for G is rainbow k-connected if any two vertices of G are connected by k internally vertex-disjoint rainbow paths. Shinya Fujita, Henry Liu ∗ , Colton Magnant Rainbow k -connection in Dense Graphs

Introduction Graphs with Fixed Connectivity Complete Bipartite and Multipartite Graphs Random Graphs Open Problems Introduction ◮ G is a finite, simple, k -connected graph ( k ∈ N ). ◮ An edge-coloured path is rainbow if its edges have distinct colours. ◮ An edge-colouring (not necessarily proper) for G is rainbow k-connected if any two vertices of G are connected by k internally vertex-disjoint rainbow paths. ◮ The rainbow k-connection number of G , denoted by rc k ( G ), is the minimum integer s such that there exists a rainbow k -connected edge-colouring of G , using s colours. Shinya Fujita, Henry Liu ∗ , Colton Magnant Rainbow k -connection in Dense Graphs

Introduction Graphs with Fixed Connectivity Complete Bipartite and Multipartite Graphs Random Graphs Open Problems Introduction ◮ G is a finite, simple, k -connected graph ( k ∈ N ). ◮ An edge-coloured path is rainbow if its edges have distinct colours. ◮ An edge-colouring (not necessarily proper) for G is rainbow k-connected if any two vertices of G are connected by k internally vertex-disjoint rainbow paths. ◮ The rainbow k-connection number of G , denoted by rc k ( G ), is the minimum integer s such that there exists a rainbow k -connected edge-colouring of G , using s colours. ◮ Write rc( G ) = rc 1 ( G ). Shinya Fujita, Henry Liu ∗ , Colton Magnant Rainbow k -connection in Dense Graphs

Introduction Graphs with Fixed Connectivity Complete Bipartite and Multipartite Graphs Random Graphs Open Problems Introduction ◮ G is a finite, simple, k -connected graph ( k ∈ N ). ◮ An edge-coloured path is rainbow if its edges have distinct colours. ◮ An edge-colouring (not necessarily proper) for G is rainbow k-connected if any two vertices of G are connected by k internally vertex-disjoint rainbow paths. ◮ The rainbow k-connection number of G , denoted by rc k ( G ), is the minimum integer s such that there exists a rainbow k -connected edge-colouring of G , using s colours. ◮ Write rc( G ) = rc 1 ( G ). ◮ Note: rc k ( G ) is well-defined if G is k -connected (by Menger’s Theorem). Shinya Fujita, Henry Liu ∗ , Colton Magnant Rainbow k -connection in Dense Graphs

Introduction Graphs with Fixed Connectivity Complete Bipartite and Multipartite Graphs Random Graphs Open Problems Example G = C 5 + v , the wheel with five spokes. Shinya Fujita, Henry Liu ∗ , Colton Magnant Rainbow k -connection in Dense Graphs

Introduction Graphs with Fixed Connectivity Complete Bipartite and Multipartite Graphs Random Graphs Open Problems Example G = C 5 + v , the wheel with five spokes. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Shinya Fujita, Henry Liu ∗ , Colton Magnant Rainbow k -connection in Dense Graphs