Labelling graphs with a condition at distance 2
SIAM Journal on Discrete Mathematics
On L(d, 1)-labelings of graphs
Discrete Mathematics
A coloring problem on the n-cube
Discrete Applied Mathematics
Fixed parameter complexity of λ-labelings
Discrete Applied Mathematics - special issue on the 25th international workshop on graph theoretic concepts in computer science (WG'99)
New bounds on a hypercube coloring problem
Information Processing Letters
Distance Constrained Labeling of Precolored Trees
ICTCS '01 Proceedings of the 7th Italian Conference on Theoretical Computer Science
Labeling trees with a condition at distance two
Discrete Mathematics
A channel assignment problem for optical networks modelled by Cayley graphs
Theoretical Computer Science
Coloring Powers of Chordal Graphs
SIAM Journal on Discrete Mathematics
Labelling Cayley Graphs on Abelian Groups
SIAM Journal on Discrete Mathematics
The L(h, k)-Labelling Problem: A Survey and Annotated Bibliography
The Computer Journal
Computational Complexity of the Distance Constrained Labeling Problem for Trees (Extended Abstract)
ICALP '08 Proceedings of the 35th international colloquium on Automata, Languages and Programming, Part I
A distance-labelling problem for hypercubes
Discrete Applied Mathematics
L(h,1,1)-Labeling of outerplanar graphs
SIROCCO'06 Proceedings of the 13th international conference on Structural Information and Communication Complexity
Distance constrained labelings of graphs of bounded treewidth
ICALP'05 Proceedings of the 32nd international conference on Automata, Languages and Programming
Distance three labelings of trees
Discrete Applied Mathematics
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Let h=1 be an integer. An L(h,1,1)-labelling of a (finite or infinite) graph is an assignment of nonnegative integers (labels) to its vertices such that adjacent vertices receive labels with difference at least h, and vertices distance 2 or 3 apart receive distinct labels. The span of such a labelling is the difference between the maximum and minimum labels used, and the minimum span over all L(h,1,1)-labellings is called the @l"h","1","1-number of the graph. We prove that, for any integer h=1 and any finite tree T of diameter at least 3 or infinite tree T of finite maximum degree, max{max"u"v"@?"E"("T")min{d(u),d(v)}+h-1,@D"2(T)-1}@?@l"h","1","1(T)@?@D"2(T)+h-1, and both lower and upper bounds are attainable, where @D"2(T) is the maximum total degree of two adjacent vertices. Moreover, if h is less than or equal to the minimum degree of a non-pendant vertex of T, then @l"h","1","1(T)@?@D"2(T)+h-2. In particular, @D"2(T)-1@?@l"2","1","1(T)@?@D"2(T). Furthermore, if T is a caterpillar and h=2, then max{max"u"v"@?"E"("T")min{d(u),d(v)}+h-1,@D"2(T)-1}@?@l"h","1","1(T)@?@D"2(T)+h-2 with both lower and upper bounds achievable.