yu-jung liang department of applied mathematics national dong hwa university advisor: professor...
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Rainbow connection numbers of Cartesian product of graphs
Yu-Jung Liang
Department of Applied MathematicsNational Dong Hwa University
Advisor: Professor David Kuo
Outline
Introduction
Previous Results
Main Results
Definition (The Cartesian product)Give two graphs and , the Cartesian product
of and , denoted by , is defined as follows: .Two distinct vertices and of are adjacent if and only if either and or and .
Example:
Definition (Rainbow connection number)A path is rainbow if no two edges of it are
colored the same.An edge-coloring graph is rainbow
connected if any two vertices are connected by a rainbow path.
We define the rainbow connection number of a connected graph , denoted by rc, as the smallest number of colors that are needed in order to make rainbow connected.
A graph is strong rainbow connected if there exists a rainbow geodesic for any two vertices and in .
Example:
1 2 3 4
5
2 3 4
5
1 2 3 4 51
1 2 3 1
2
2 3 1
2
1 2 3 1 21
rc
Previous ResultsTheorem 1 (Xueliang Li, Yuefang Sun)For any connected graph .Theorem 2 (Xueliang Li, Yuefang Sun)Let , where each is connected. Then we have
.Moreover, if for each , then the equality holds.
back1
Theorem 3 (M. Basavaraju, L.S. Chandran, D. Rajendraprasad, and A. Ramaswamy)
If and are non-trivial connected graphs, then .
back
Main ResultsParticular labeling
0
0
1
1
1
2
2
32
3
3
0 4
0
0
1
1
1
2
2
32
3
3
0 4
1
0
Rainbow connection numbers of the Cartesian product of paths and cyclesIf , then for all .If , then for all .If n is even, then , for all .
Example:
1 2 3 1
3
2 3 1
3
1 2 3 1 21
1 2
21 2 3
1
Thm1
Rainbow connection numbers of the Cartesian product of two trees
If and are trees, we have follows conclusion about .
If , , then .
12 3
1 2 3
1
3
Thm3
If , has three edge-disjoint path with length , then .
Otherwise, we have .Example
1 1
2 2
3 3
4 4
5
6
7
6
4 4
5 5
6 6
1 1
2
3
7
3
Example
1 1
2 2
3 3
7
4 4
5
6
8
7 7
8
4 4 1 1
2
3 3
5 5
6 6
7
8
8
88
8
8
8
8
Example
1
2
3
4
5
6 6
7 81
2
3
4
5
63
2
1
3
4
5
6
1
2
3 3
7 84
5
6
1
2
36
5
4
6
Example
1
2
3
4
5
6 6
7 8
9
1
2
3
7
8
7
9
9
9 9
9
9
9 9
9 9
9
9
9
99
7 84 4
115
66
4
5
6 6
5
22
333
Example
12
31
2
3
1
23
1
In fact, if we consider subgraph of , we have three nature graph. Hence easy to check have rainbow path for any vertices in .1
112
2
32
1
112
3
13
3
312
2
32
12
31
2
3
1
23
1
