图的点荫度和点线性荫度

马刚山东大学数学院

The vertex arboricity va(G) of a graph G is the minimum number of colors that can be used to color the vertices of G so that each color class induces a forest of G.

The vertex linear arboricity vla(G) of a graph G is the minimum number of colors that can be used to color the vertices of G so that each color class induces linear forest of G.

For any graph G, ( ) ( ) ( )vla G va G G

Theorem (Kronk and Mitchem, 1975) Let G be a simple connected graph. If G neither a cycle nor a clique of odd order, then

( )( )2

Gva G

Theorem (Matsumoto,1990) Let G be a connected graph. Then

(1)There exists a coloring of G such that each induced subgraph has only or as its connected components.

(2) .

(3)If for some positive integer n, then if and only if G is a cycle or .

( )(1 )

2

G

iG 2K 1K

( )( ) 1

2

Gvla G

( ) 2G n

2 1nK

( )( ) 1

2

Gvla G

Theorem (Akiyama, Era, Gervacio and Wtanabe, 1989) If G is a graph with maximum degree d, then

2

1( , )

2

dG P

Theorem (Catlin and Lai, 1995) Let k be a natural number and let G be a connected simple graph with that is not a complete graph (if ) nor a cycle (if k=1). Then and there is a k-coloring of G such that each color class induces a forest, and such that one color class is a maximum induced forest in G.

2k

1k ( )va G k

Theorem (Catlin and Lai, 1995) Let G be a connected simple graph ,and let k be a positive integer, then G has a (k+1)-coloring ,where each color class is a forest .Further more ,if G is not a complete graph then for each property below, this coloring can be chosen to satisfy that property:

(a) one color class is edgeless and one color class may be assumed to be a maximum induced forest, or

(b) one color class may be assumed to be a maximum independent set.

( ) 2 1G k

Theorem (Burr, 1986) For every graph G, . Moreover, for every , there is a G with va(G)=a(G)=k.

( ) ( )va G a G 1k

Theorem (Michem 1970) Let G be any graph of order p. Then

And the bounds are sharp.

1( ) ( ) 1

2

pp va G va G

23( ) ( ) [ ]4 4

p pva G va G

Theorem (Alavi, Green, Liu, Wang,1991) Let G be any graph of order p. Then

and the lower bounds are sharp except for the sum in the case .

2( ) ( ) 1

3

pp vla G vla G

22 3( ) ( ) [ ]

4 6

p pvla G vla G

2(2 1)p n

Theorem (Alavi, Liu, Wang, 1994) Let G be any graph of order p. Then

and for any graph G of order , where , ,

and all the bounds are sharp.

1( ) ( ) 1

2

pvla G vla G

2( ) ( ) ( ( 3) / 2 / 2)vla G vla G p

2(2 1)p n

n Z 2 2 ( ) ( )n vla G vla G

Theorem (Lam, Shiu, Sun, Wang, Yan, 2001) If G is a graph of order n, then

and all of the bouds are sharp.

Theorem (Lam, Shiu, Sun, Wang, Yan, 2001) If G is a graph of order n, then

and all of the bouds are sharp.

2 ( ) ( ) 1n vla G G n

2( 3)( ) ( )

2 8

n nvla G G

2 ( ) ( ) 1n vla G G n 2( 3)

( ) ( )2 8

n nvla G G

Theorem (Poh, 1990) If G is a planar graph, then

( ) 3vla G

Theorem( 杨爱民， 1998)

(1)

(2) If G is a tree , then

2( ( ))

1 2 1n

p n pva L K

p n p

( )( ( ))

2

Gva L G

Theorem ( 左连翠，吴建良，刘家壮， 2006)

(1) If and , then for an interval D between 1 and .

(2) Let , then

for

for

for

for

1n n 1n ( ( , )) 1vla G D n

, {1,..., } \{ }m kD m k

,1( ( )) 14m

mvla G D

,2

11 ( ( )) 2

4 4m

m mvla G D

,( ( ))2m k

kvla G D

,

1( ( )) ( 1)

4 4m k

m k mvla G D k

k

3m

6m

12

km k

3m k

Theorem （马刚，吴建良， 2006 ） (1)If T is a tree with maximum

degree ， then

(2)If T is a tree with maximum degree ， then

(3) If G is an outerplanar graph with maximum degree ， then

2 1( )

2va T

21( ) 1

2 2vla T

6

2 1( )

2va G

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