摘要
An edge-cut of an edge-colored connected graph is called a rainbow cut if no two edges in the edge-cut are colored the same.An edge-colored graph is rainbow disconnected if for any two distinct vertices u and v of the graph,there exists a rainbow cut separating u and v.For a connected graph G,the rainbow disconnection number of G,denoted by rd(G),is defined as the smallest number of colors required to make G rainbow disconnected.In this paper,we first give some upper bounds for rd(G),and moreover,we completely characterize the graphs which meet the upper bounds of the NordhausGaddum type result obtained early by us.Secondly,we propose a conjecture that for any connected graph G,either rd(G)=λ^(+)(G)or rd(G)=λ^(+)(G)+1,whereλ^(+)(G)is the upper edge-connectivity,and prove that the conjecture holds for many classes of graphs,which supports this conjecture.Moreover,we prove that for an odd integer k,if G is a k-edge-connected k-regular graph,thenχ’(G)=k if and only if rd(G)=k.It implies that there are infinitely many k-edge-connected k-regular graphs G for which rd(G)=λ^(+)(G)for odd k,and also there are infinitely many k-edge-connected k-regular graphs G for which rd(G)=λ^(+)(G)+1 for odd k.For k=3,the result gives rise to an interesting result,which is equivalent to the famous Four-Color Problem.Finally,we give the relationship between rd(G)of a graph G and the rainbow vertex-disconnection number rvd(L(G))of the line graph L(G)of G.
基金
Supported by National Natural Science Foundation of China(Grant No.11871034)。
