Rainbow connection of graphs with diameter 2

Li, Hengzhe; Li, Xueliang; Liu, Sujuan

doi:10.1016/j.disc.2012.01.009

Cited by 37 publications

(34 citation statements)

References 9 publications

Supporting

Mentioning

Contrasting

Unclassified

Order By: Relevance

“…In [43], Dong and Li gave another proof for the upper bound 5, and moreover, examples are given to show that the bound is best possible. In [84], Li et al also showed that rc(G) ≤ k + 2 if G is connected with diameter 2 and k bridges, where k ≥ 1. The bound k + 2 is sharp as there are infinity many graphs of diameter 2 and k bridges whose rainbow connection numbers attain this bound.…”

Section: Rainbow Connection Coloring Of Edge-versionmentioning

confidence: 99%

See 1 more Smart Citation

An Updated Survey on Rainbow Connections of Graphs- A Dynamic Survey

Li¹,

Sun²

2017

tag

Self Cite

View full text Add to dashboard Cite

The concept of rainbow connection was introduced by Chartrand, Johns, McKeon and Zhang in 2008. Nowadays it has become a new and active subject in graph theory. There is a book on this topic by Li and Sun in 2012, and a survey paper by Li, Shi and Sun in 2013. More and more researchers are working in this field, and many new papers have been published in journals. In this survey we attempt to bring together most of the new results and papers that deal with this topic. We begin with an introduction, and then try to organize the work into the following categories, rainbow connection coloring of edge-version, rainbow connection coloring of vertex-version, rainbow k-connectivity, rainbow index, rainbow connection coloring of total-version, rainbow connection on digraphs, rainbow connection on hypergraphs. This survey also contains some conjectures, open problems and questions for further study.

show abstract

Section: Rainbow Connection Coloring Of Edge-versionmentioning

confidence: 99%

“…In [84], Li et al showed that rc(G) ≤ 5 and they also gave examples for which rc(G) ≤ 4. However, they could not show that the upper bound 5 is sharp.…”

Section: Rainbow Connection Coloring Of Edge-versionmentioning

confidence: 99%

An Updated Survey on Rainbow Connections of Graphs- A Dynamic Survey

Li¹,

Sun²

2017

tag

Self Cite

View full text Add to dashboard Cite

show abstract

“…Given the edge-coloring in the proof of Lemma 4.1 of [10], we can obtain a T RCcoloring of a bridgeless graph G of diameter 2 with a cut-vertex v by assigning 4 to the vertex v and 5 to the other vertices of G. 10]). If G is a 2-connected graph with diameter 2, then rc(G) ≤ 5.…”

Section: Upper Bound On Trc(g) + Trc(g)mentioning

confidence: 99%

Tight Nordhaus–Gaddum-Type Upper Bound for Total-Rainbow Connection Number of Graphs

Magnant

et al. 2017

Results Math

Self Cite

View full text Add to dashboard Cite

A graph is said to be total-colored if all the edges and the vertices of the graph are colored. A total-colored graph is total-rainbow connected if any two vertices of the graph are connected by a path whose edges and internal vertices have distinct colors. For a connected graph G, the total-rainbow connection number of G, denoted by trc(G), is the minimum number of colors required in a total-coloring of G to make G total-rainbow connected. In this paper, we first characterize the graphs having large total-rainbow connection numbers. Based on this, we obtain a Nordhaus-Gaddum-type upper bound for the total-rainbow connection number. We prove that if G and G are connected complementary graphs on n vertices, then trc(G) + trc(G) ≤ 2n when n ≥ 6 and trc(G) + trc (

show abstract

“…In [14], Li et al also showed that rcðGÞ 6 k þ 2 if G is connected with diameter 2 and k bridges, where k P 1. The bound k þ 2 is sharp as there are infinity many graphs with diameter 2 and k bridges whose rainbow connection numbers attain this bound.…”

Section: Introductionmentioning

confidence: 96%

“…So, graphs with rcðGÞ ¼ 2 belong to the graph class with diamðGÞ ¼ 2. Therefore, there is an interesting problem: For any bridgeless graph G with diamðGÞ ¼ 2, determine the smallest constant c such that rcðGÞ 6 c. In [6,14], Li et al showed that c 6 5, and moreover, examples are given to show that the bound is best possible.…”

Section: Introductionmentioning

confidence: 98%