图的点荫度和点线性荫度

1. 图的点荫度和点线性荫度 马刚 山东大学数学院

2. 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,

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

4. 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 .

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

6. 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.

7. 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.

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

9. Theorem (Michem 1970) Let G be any graph of order p. Then And the bounds are sharp.

10. 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 .

11. 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.

12. 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.

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

14. Theorem(杨爱民，1998) (1) (2) If G is a tree , then

15. 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

16. 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

17. J. Akiyama, H. Era, S.V. Gervacio, and M. Vatanabe, Path chromatic numbers of graphs. J. Graph Theory, 13 (1989) 569-575. • Y. Alavi, D. Green, J. Q. Liu and J. F. Wang, On linear vetex-arboricity of graphs. Congressus Numeratim, 1991, 82, 187-192 • Y. Alavi, P. C.B. Lam, D. R. Lick, P. Erdos, J. Q. Liu and J. F. Wang, Upper Bounds on linear-vertex arboricity of complementary Graphs. Utilitas Math. 52, (1997) 43-48. • Y. Alavi, J. Q. Liu and J. F. Wang, On linear vertex-arboricity of complementary graphs. J. Graph Theory 18 (1994) 315-322. • Y. Alavi, D. R. Lick, J. Q. Liu and J. F. Wang, Bounds for linear vertex-arboricity and domination number of graphs. Vishna International Journal of Graph Theory, 1992, 1 (2), 95-102. • S. A. Burr, An inequality involving the vertex arboricity and edge arboricity of a graph . J.Graph Theory 10 (1986) 403-404. • P. A. Catlin and H. J. Lai, Vertex arboricity and maximum degree. Dis. Math. 141 (1995) 37-46. • G. Chartrand, D. P. Geller, and S. Hedetniemi, A generalization of the chromatic number . Proc. Camb. Phil. Soc. 64 (1968) 265-271.

18. G. Chartrand and H. V. Kronk, The point-arboricity of planar graphs. J. London Math. Soc. 44 (1969), 612-616. • G. Chartrand, H. V. Kronk and C. E. Wall, The point-arboricity of a graph. Israel J. Math. 6 (1968).169-175. • F. Harary, R. Maddox and W. Staton. On the point linear arboricity of a graph. Mathematiche (Catania) 44 (1989) 281-286. • H. V. Kronk and J. Mitchem, Critical point-arboritic graphs. J. London Math. Sco. (2) ,9 (1975), 459-466. • P. C. B. Lam, W. C. Shiu, F. Sun, J. F. Wang and G. Y. Yan, Linear vertex arboricity, independence number and clique cover number. Ars Combin. 58 (2001) 121-128. • M. Matsumoto, Bounds for the vertex linear arboricity. J. Graph Theory 14 (1990) 117-126. • J. Mitchem, Doctoral thesis, Western Michigan University(1970). • K. S. Poh, On the linear vertex-arboricity of a planar graph. J. Graph Theory, 14 (1990) 73-75. • J. F. Wang, On point-linear arboricity of planar graphs. Discrete Math. 72 (1988) 381-384.

19. L. C. Zuo, J. L. Wu, J. Z. Liu, The vertex linear arboricity of an integer distance graph with a special distance set. Ars Combin. 79 (2) (2006), preprint. • L. C. Zuo, J. L. Wu, J. Z. Liu, The vertex linear arboricity of distance graphs. Discrete Math. 306 (2006) 284-289. • 房勇，吴建良，完全多部图和笛卡儿乘积图的线性点荫度. 山东矿业学院学报（自然科学版） 18（3），59-61，1999. • 杨爱民，线图的荫度. 山西大学学报（自然科学版）21（1）：19-22，1998. • 张忠辅, 王建方, 图与补图全独立数间的关系。应用数学，1989，2（4）35-39. • 张忠辅，张建勋，王建方，陈波亮，荫度与全独立数全覆盖数的关系. 曲阜师范大学学报 18（4），9-14，1992. • 左连翠，李涛，李霞，整数距离图的点线性荫度. 山东大学学报（理学版）39（6），68-71，2004. • 左连翠，李霞，距离图的点荫度. 山东大学学报（理学版）39（2），12-15，2004.

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