The Conditional Coloring And L (2, 1)-labeling Of Generalized Petersen Graphs

The study on graph colorings and labelings are very important and closely related research topicss in graph theory. They have strong backgrounds and direct applications both in theory and in engineering applications and real applications. A lot of technology, management, industrial design and other fields can be formulated as graph colorings and labelings problems to solve, such as task scheduling, scheduling problem, signal frequency assignment problems, storage problems, resource allocation. VLSI wiring and so on. In this paper. conditional coloring and L(2.1)-labeling of generalized Petersen graphs are discussed.The concept of conditional coloring proposed by Lai Hongjian in 2001. In this paper. conditional coloring of generalized Petersen graphs is discussed.and we proved that if a graph G is a generalized Petersen graph P(n.t). but it isn't Petersen graph P(5.2), (1) if(?)= 5. then X3(G)= 5;(2) if(?), then X3(G)≤6:(3) otherwise,X3(G)≤7.The L(2.1)-labeling is introdured by Gringgs and Yeh in 1992. So far. there are many results of L(2.1)-labeling studies. In this paper,L(2.1)-labeling of generalized Petersen graphs is discussed, we proved that if a graph G is a generalized Petersen graph P(n.t). but it isn't Petersen graph p(5.2), (1)for(?)0(mod3). if d≠0(mod3). then (?)(G)= 5; (2)for (?)≠0(mod3),d≡(mod3), if d=2, then(?)(G)≤10; if d≥5, then (?)(G)<9; (3) otherwise.(?)(G)≤8.In addition, we give the upper bound of 6 and 7 of the generalized Petersen graphs.