Labelling graphs with a condition at distance 2
SIAM Journal on Discrete Mathematics
The $L(2,1)$-Labeling Problem on Graphs
SIAM Journal on Discrete Mathematics
On L(d, 1)-labelings of graphs
Discrete Mathematics
Labeling trees with a condition at distance two
Discrete Mathematics
L(h,1)-labeling subclasses of planar graphs
Journal of Parallel and Distributed Computing
List version of L(d, s)-labelings
Theoretical Computer Science - Graph colorings
The L(2, 1)-labelling of trees
Discrete Applied Mathematics
Distance constrained labelings of graphs of bounded treewidth
ICALP'05 Proceedings of the 32nd international conference on Automata, Languages and Programming
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Given a graph G = (V, E) and a positive integer d, an L(d,1)-labelling of G is a function f : V (G) → {0, 1, &} suchthat if two vertices x and y are adjacent, then ' f (x) − f(y) ' ≥ d; if they are at distance 2, then ' f (x) − f (y)' ≤ 1. The L(d, 1)-number of G, denoted by λd,1(G), is the smallest number m such that G has an L(d, 1)-labellingwith max{f (x) : x ∈ V (G)} = m. It is known that for diameter2 graphs it holds λ2,1 ≤ Δ2.We will prove Δ2+(d−2)Δ−1 as anupper bound for λd,1 for all diameter 2 graphs oforder ≤ Δ2 − 1 and d ≥ 2. For diameter2 graphs of maximum degree Δ = 2, 3, 7 and orderΔ² + 1 exact values or tight bounds forλd,1 will be presented. After this we provide alower bound for λd,1 for trees. At last an upperbound for λd,1 for cacti will be established. Inparticular, we determine λ1,1 for all cacti andwe will show λd,1 ≤ Δ + 2d − 2 forcacti having large girth or a large maximum degree Δ.