L(j,k)-labeling is a kind of generalization of the classical graph coloring motivated from a kind of frequency assignment problem in radio networks, in which adjacent vertices are assigned integers that are at least j apart, while vertices that are at distance two are assigned integers that are at least k apart. The span of an L(j,k)-labeling of a graph G is the difference between the maximum and the minimum integers assigned to its vertices. The L(j,k)-labeling number of G, denoted by @l"j","k(G), is the minimum span over all L(j,k)-labelings of G. Georges, Mauro and Whittlesey (1994) [1] established the relationship between @l"2","1(G) of a graph G and the path covering number of G^c (the complement of G). Georges, Mauro and Stein (2000) [2] determined the L(j,k)-labeling numbers of Cartesian products of two complete graphs. Lam, Lin and Wu (2007) [3] determined the @l"j","k-numbers of direct products of two complete graphs. In 2011, we (Wang and Lin, 2011 [4]) generalized the concept of the path covering to the t-group path covering of a graph where t(>=1) is an integer and established the relationship between the L^'(d,1)-labeling number (d>=2) of a graph G and the (d-1)-group path covering number of G^c. In this paper, we establish the relationship between the @l"j","k(G) of a graph G with diameter 2 and the @?j/[email protected]?-group path coverings of G^c. Using those results, we can have shorter proofs to obtain the @l"j","k of the Cartesian products and direct products of complete graphs.
[1]
Jerrold R. Griggs,et al.
Labelling Graphs with a Condition at Distance 2
,
1992,
SIAM J. Discret. Math..
[2]
W. K. Hale.
Frequency assignment: Theory and applications
,
1980,
Proceedings of the IEEE.
[3]
John P. Georges,et al.
Relating path coverings to vertex labellings with a condition at distance two
,
1994,
Discret. Math..
[4]
Daphne Der-Fen Liu.
Hamiltonicity and circular distance two labellings
,
2001,
Discret. Math..
[5]
John P. Georges,et al.
Labeling Products of Complete Graphs with a Condition at Distance Two
,
2001,
SIAM J. Discret. Math..
[6]
Peter Che Bor Lam,et al.
L(j,k)- and circular L(j,k)-labellings for the products of complete graphs
,
2007,
J. Comb. Optim..
[7]
Wensong Lin,et al.
Group path covering and distance two labeling of graphs
,
2011,
Inf. Process. Lett..
[8]
Daphne Der-Fen Liu,et al.
On L(d, 1)-labelings of graphs
,
2000,
Discret. Math..
[9]
John P. Georges,et al.
Labeling trees with a condition at distance two
,
2003,
Discret. Math..
[10]
G. G. Stokes.
"J."
,
1890,
The New Yale Book of Quotations.