The Packing Coloring and the (p,1) Total Labelling of Graphs 

Author  ZhangZuoZuo 
Tutor  SunLei 
School  Shandong Normal University 
Course  Applied Mathematics 
Keywords  (p, 1)  total labeling Pan width dyeing (p, 1)  the number of total labeling Pan width color number Cross Fig. Complete bipartite graph Cartesian product hajós sum Split Graphs 
CLC  O157.5 
Type  Master's thesis 
Year  2009 
Downloads  21 
Quotes  0 
Graph theory is a very young discipline, but mature quickly, it is a kind of in various scientific fields like computer science, physics, biology, chemistry, strategic science disciplines in the application model. Graph coloring problem is a diagram of the in one of the main areas of the frequency allocation problem (The Prequency Assignment Problem (FAP)) is a general structure to focus on the problem of peertopeer communication, such as radio communication and mobile phone networks. frequency requirements used to spread or interference between channels is maintained at an acceptable level, and thus the effective use of the available frequency in this issue, we need to allocate frequency band to the transit station in order to avoid interference from the two sites are very close, then they frequency difference of at least 2. Moreover, if the two sites from near (not very close), then the frequency that they are inspired by this problem, Griggs and Yeh [10] introduced the L (2,1)  label .2000, GJChang et al [11] to extend it to the L (p, 1)  label. G, three (p, 1)  label is an integer assigned set of vertices of G such that: if d _{ G } (u, v) = 1, then  L (u)L (v)  ≥ P; if D to _{ a G } (u, v) = 2, then  L (u)L (v) 1 ≥ 1. the label in the number of articles has been studied before, [11] studied the chord chart Whittesey et al [12] studied the subdivision graph of G L (2,1)  label. subdivision graph of G S _{ 1 } (G), insert a point Figure by G, each edge. s _{ 1 (G) of L (p, 1)  label corresponds to on the original graph G (p, 1)  total labeling [13]: Let p be a nonnegative integer, Figure G is a k(p, 1 )  Full numeral is a mapping f: V (G) ∪ E (G) → {0,1, ..., k) such that: (1) G of any two adjacent vertices u and v have  F ( u)f (u)  ≥ 1; (2) G of any two adjacent edges e, e ', there  f (E)f (e')  ≥ 1; any of (3) G two related points u and edge e,  f (u)f (e)  ≥ p we call such a distribution called G, (p)  total labeling. (p, 1)  full span of the label refers to the the two labels difference in maximum of G (p, 1)  the smallest span of the whole label called (p, 1)  total labeling number recorded as the λ p < sup> T (G). it is a way of strengthening the dyeing conditions, the additional conditions associated point and the edge of the label to be a difference of at least noticed (1,1)  total labeling is the total coloring , λ 1 T = χ T 1, where χ T is a fullcolor number in the literature [ 22] gives the following definitions: Definition 1 [22] , G = (V (G), E (G)), d is a positive integer, X is V (G) of a subset, if x is the distance of any two points is greater than d, then X is the width of the dbox, d the width of the called X Obviously, dwidth of the box is (d1)  width of the box, (d 2)  width box and so on. defined 2 [22] Given a graph G = (V (G), E (G)), such that V (G) = (?), and X i (i = 1 ... k) is the smallest integer k pairwise distinct V (G) the width of the iwidth box called the pan width of the graph G number of colors, referred to as χ ρ (G). χ ρ (G), staining, a size of the graph G called χ the ρ (G)  staining. Obviously, here we The purpose is to minimize k can assume that for each i, the set is an iwidth box. panwidth dyeing requirements all vertices are partitioned into the width of the twotwo different the iWidth box X i the the same width box X i, , the distance between any two points is greater than i, X , i of, vertex requires the use of a Pan width staining is a special point of the same color. staining, the past two years, began its study of the concept of channel allocation, the allocation of resources and biodiversity issues in many applications this paper, we get the following conclusions: definition 2.1.1 [26] < / sup> B = {B 1 , B 2 , ..., B n ) is a family of sets we call n vertices Figure (B)) = {v i v j } (?), j = 1,2, ..., n; B i ∩ B j ≠ 0.}. set up the Z n = {1,2, ..., n}, its power set Z = {Z 1 , Z 2 , ..., Z 2 n  (?)). we consider the family of sets (?) consisting of crossFigure X ( Z ') and (p, 1)  total labeling. Theorem 2.1.1 Exchange Figure X (Z') as described above, Theorem 2.1.2 If K n, m (m> n ≥ 1) complete bipartite graph, and △> p (1) n = 1, m ≥ 2, λ p T (K n < / sub>, m) = M p1 (2) n ≥ 2, m> n, (i) if the meet Mn ≥ 2p1, then λ p T < / sup> (K n , m) = m p1. (ii) is satisfied mn ≥ p mn <2p1, then λ p T (K n , m) = m p. Lemma 2.2.1 [20] G satisfies the λ p <. sup> T (G) = △ (G) 1 and a (△ (G) 1)  (2,1)  total labeling: y (G) ∪ E (G) → { 1,2, ..., △ (G) 1), the each maximum degree point v, f (v) = 0 or f (v) = △ (G). the lemma 2.2.2 If G satisfies λ < sub> p T (G) = △ (G) p1 and f is a (△ (G) p1)  (p, 1)  total labeling: V (G) ∪ E (G) → (0,1,2, ..., △ (G) p1), each of the maximum degree point v, f (v) = 0 or F (v) = △ ( G) p1. Theorem 2.2.3 △> p, if G contains a maximum degree of a vertex v is adjacent to at least d (v)p maximum degree point, and the maximum degree of points generated sub graphs without triangles , then λ p T (G) ≥ △ p. of definition 2.2.1 G simple graph, V (G) = {v 1 v 2 , ..., the v n }, the vertex v m a G j 0 ∈ G ( j 0 ∈ {1,2, ..., n)) even into a circle, the new graph, denoted as the C m · G (v j < sub> 0 }), abbreviated as G ^{ * G i (i = 1,2, ..., m) for mth G copy Fig. namely a set of points and the set of edges of the new graph G * : Theorem 2.2.4 G with maximum degree △ (G), and the λ p T (G) = △ (G) p1, take v j 0 G of the maximum degree of any point, wish denoted v 1 definition 2.2.1 new stamp for the C m · G (v 1 ), abbreviated as G 1 < / sub> * . Theorem 2.2.5 take G meet: of G (p, 1)  total labeling number 0 ≤ λ , p , T (G) ≤ 2p1, Let f G a (p, 1)  total labeling: f: V (G) ∪ E (G) → [0, 1, ..., λ p j 0 v 0 may wish to take note of is v 1 Illustrated by definition 2.2.1 C < sub> m · G (v 1 ),, abbreviated G 2 * . Suppose that G v 1 adjacent sides of the maximum designated PA (1 ≤ a ≤ λ p T (G)p), then m is an even number, m is odd, defined 2.2.2 T as a tree,  T  = m, V (T) = {U 1 , u 2 , ..., u m }, except leaves and T is equal to the degree of the remaining points of all the points, △ (T) = △ G is an norder view, V (G) = {V L < / sub> v 2 , ..., v n ), if G is (p, 1)  total labeling number λ p < sup> T (G), G △ (△ ≥ 2) label the same point v 1 , v 2 , ..., v △ , its label is set to a 0 (0 ≤ a 0 ≤ the λ p T (G)), and with each V i (i = 1,2, ..., △) adjacent the reference numerals of all of the edges are less than adjacent v i label for G m a copy Figure of G 1 , G 2 , ..., G m , G i (i = 1,2, ..., m) instead of u i ({= 1,2, ..., m), when G i , G < sub> j points in the corresponding T u i , u j adjacent G i a labeled a 0 point G j , a label for a 0 of connect the dots, and G i (i = 1,2, ..., m) in the reference numeral for a 0 point only with G j (where j ≠ i) in a reference numeral for a < sub> 0 of the point adjacent to the point G of the remaining edge  the same way. The thus obtained Illustrated T · G (V 1 , v 2 < / sub>, ..., v △ ), abbreviated as G ' T Obviously this figure G' T is not the only may not be isomorphic. but any one obtained in this manner FIG following theorem is established. the Theorems 2.2.6 G ' T As described above, the λ p T (G) ≤ λ p T (G ' T ) ≤ λ p T (G) 2. definition 2.2.2 G △ same label point v 1 v 2 , ..., v △ , conditions were changed to the right (?) (i = 1,2, ..., △), at least one edge in the adjacent side of e numeral greater than therewith reference numerals of adjacent v i , and all v 1 , v 2 , ..., V △ adjacent the edges labeled pa (0 ≤ a ≤ λ p T (G)p). construct a new map, so that the structure in accordance with the definition 2.2.2 Illustrated obtained T · G (v 1 , v 2 , ..., V △ ), abbreviated as G \T . obviously thus obtained FIG G T is not the only, but also may not be isomorphic. FIG any one obtained in this manner the following theorem is established . Theorem 2.2.7 G \H is defined as follows: V (G □ H) = V (G) x V (H); E (G □ H) = {(u, v) (u'v ')  V = v' and UU '∈ E (G) or u = u 'and VV' ∈ E (H)). Theorem 2.3.1 P m for the length of the way of m, the V (of P m ) = {u 1 , u 2 , ..., u m , u m 1 }. H meet:  H  = n, V (H) = {V 1 , the V 2 , ..., the V n }, H (p, 1 )  total labeling number λ p T (H), and there is a point v and its adjacent sides, at least one side of the label label greater than v reset all largest label in the side of the pa (a ≥ 0), the for P m H of the Cartesian product P m □ H, V (P , m □ H) = {(u i , v j )  i = 1,2, ..., m 1; j = 1,2, ..., n) ), is defined in, 3.1.1 [51] G is a simple graph, V (G) = {v 1 v 2 , ... independent sets of two points, v n }, I 2 , G [I 2 ] is 2 instead of each vertex of G Figure G [I 2 ] point set and edge set as follows: G [I 2 ] point called G split in Fig. defined by the split graph is given some special split pan width color, defines a new class diagram based on the wheel, and such plans and split the Panwidth color number Definitions 3.2.1 [36] given the m Figure G 0 , G 1 , ..., G m1 , i = 0,1, ..., m1, ORDER the E i = the V i the V i 1 ∈ G i , so the c m = c 0 c 1 ... c m1 m for long the circle, construct new plan: (1) by deleting e i , i = 0,1, ..., m1 (2) merge all the v i < / sub> into a point x (3) to v i 1 with c i Synthesis of a point, i = 0,1, ..., m1 (I 1 modulo m) This picture shows the S 1 (G 0 e 0 , G 1 e 1 , ..., G , m1 the e m1 ), abbreviated as S 1 . S given the m Figure K n i , i = 1,2, ..., m, 1 (G 0 < / sub>, e 0 , G 1 , e 1 , ..., G m1 , e < sub> m1 ) = S 1 (K n 1 , e 1 , K n 2 , e 2 , ..., K n m , e m ), abbreviated as S ' 2 . Vocabulary the all v i merged into a point x is v the 0 , v i 1 and c i merged into points in turn credited as v 1 v 2 , ... v m ; v i corresponding K n i in the rest of the points are denoted as v i1 , v i2 , ..., v i (n i 2) , i = 1,2, ..., m . the theorem 3.2.1S ' 2 As mentioned above, the χ ρ (S' 2 ) = (?) }