On some variants of the bandwidth minimization problem
SIAM Journal on Computing
Well-Spaced Labelings of Points in Rectangular Grids
SIAM Journal on Discrete Mathematics
A survey of graph layout problems
ACM Computing Surveys (CSUR)
Antibandwidth and cyclic antibandwidth of Hamming graphs
Discrete Applied Mathematics
| Hi-index | 5.23 |
Let G be a graph on n vertices. Given a bijection f:V(G)-{1,2,...,n}, let |f|=min{|f(u)-f(v)|:uv@?E(G)}. The separation numbers(G) (also known as antibandwidth [T. Calamoneri, A. Massini, L. Torok, I. Vrt'o, Antibandwidth of Complete k-ary trees, Electronic Notes in Discrete Mathematics 24 (2006), 259-266; A. Raspaud, H. Schroder, O. Sykora, L. Torok, I. Vrt'o, Antibandwidth and cyclic antibandwidth of meshes and hypercubes, Discrete Mathematics 309 (2009) 3541-3552] of G is then max{|f|} over all such bijections f of G. We study the case when G is a forest, obtaining the following results. 1.Let F be a forest in which each component is a star. Then s(F)=n-@m2, where @m is the minimum value of @?X|-|Y@? over all bipartitions (X,Y) of F. 2.Let d be the maximum degree of a tree T on n vertices. Then (a)s(T)=n2-c"1nd, and (b)s(T)=n2-c"2d^2log"dn, where c"1 and c"2 are absolute constants. We give constructions showing that the bound (a) is asymptotically tight when d is in the range n^1^3=4 is an absolute constant. We also show that for h=3 and odd d=3, we have s(T"h^d)=n2-@Q(d^2+dh), where T"h^d is the symmetric d-ary tree of height h, improving the estimates obtained in the first of the above-mentioned references.