Distance-two labelings of digraphs

For positive integers $j\ge k$, an $L(j,k)$-labeling of a digraph $D$ is a function $f$ from $V(D)$ into the set of nonnegative integers such that $|f(x)-f(y)|\ge j$ if $x$ is adjacent to $y$ in $D$ and $|f(x)-f(y)|\ge k$ if $x$ is of distant two to $y$ in $D$. Elements of the image of $f$ are called labels. The $L(j,k)$-labeling problem is to determine the $\vec{\lambda}_{j,k}$-number $\vec{\lambda}_{j,k}(D)$ of a digraph $D$, which is the minimum of the maximum label used in an $L(j,k)$-labeling of $D$. This paper studies $\vec{\lambda}_{j,k}$- numbers of digraphs. In particular, we determine $\vec{\lambda}_{j,k}$- numbers of digraphs whose longest dipath is of length at most 2, and $\vec{\lambda}_{j,k}$-numbers of ditrees having dipaths of length 4. We also give bounds for $\vec{\lambda}_{j,k}$-numbers of bipartite digraphs whose longest dipath is of length 3. Finally, we present a linear-time algorithm for determining $\vec{\lambda}_{j,1}$-numbers of ditrees whose longest dipath is of length 3.Comment: 12 pages; presented in SIAM Coference on Discrete Mathematics, June 13-16, 2004, Loews Vanderbilt Plaza Hotel, Nashville, TN, US

L(2,1)-labeling of oriented planar graphs

The L(2, 1)-labeling of a digraph D is a function l from the vertex set of D to the set of all nonnegative integers such that vertical bar l(x) - l(y)vertical bar >= 2 if x and y are at distance 1, and l(x) not equal l(y) if x and y are at distance 2, where the distance from vertex x to vertex y is the length of a shortest dipath from x to y. The minimum over all the L(2, 1)-labelings of D of the maximum used label is denoted (lambda) over right arrow (D). If C is a class of digraphs, the maximum (lambda) over right arrow (D), over all D is an element of C is denoted (lambda) over right arrow (C). In this paper we study the L(2, 1)-labeling problem on oriented planar graphs providing some upper bounds on (lambda) over right arrow. Then we focus on some specific subclasses of oriented planar graphs, improving the previous general bounds. Namely, for oriented prisms we compute the exact value of (lambda) over right arrow, while for oriented Halin graphs and cacti we provide very close upper and lower bounds for (lambda) over right arrow. (c) 2012 Elsevier B.V. All rights reserved

L(2,1)-labeling of oriented planar graphs

The L(2, 1)-labeling of a digraph D is a function l from the vertex set of D to the set of all nonnegative integers such that vertical bar l(x) - l(y)vertical bar >= 2 if x and y are at distance 1, and l(x) not equal l(y) if x and y are at distance 2, where the distance from vertex x to vertex y is the length of a shortest dipath from x to y. The minimum over all the L(2, 1)-labelings of D of the maximum used label is denoted (lambda) over right arrow (D). If C is a class of digraphs, the maximum (lambda) over right arrow (D), over all D is an element of C is denoted (lambda) over right arrow (C). In this paper we study the L(2, 1)-labeling problem on oriented planar graphs providing some upper bounds on (lambda) over right arrow. Then we focus on some specific subclasses of oriented planar graphs, improving the previous general bounds. Namely, for oriented prisms we compute the exact value of (lambda) over right arrow, while for oriented Halin graphs and cacti we provide very close upper and lower bounds for (lambda) over right arrow. (c) 2012 Elsevier B.V. All rights reserved

Distance-two labelings of digraphs

For positive integers j ≥ k, an L(j, k)-labeling of a digraph D is a function f from V (D) into the set of nonnegative integers such that |f(x) − f(y)| ≥ j if x is adjacent to y in D & |f(x) − f(y)| ≥ k if x is of distant two to y in D. Elements of the image of f are called labels. The L(j, k)-labeling problem is to determine the ~λj,k- number ~λj,k(D) of a digraph D, which is the minimum of the maximum label used in an L(j, k)-labeling of D. This paper studies ~λj,k-numbers of digraphs. In particular, we determine ~λj,k-numbers of digraphs whose longest dipath is of length at most 2, and ~λj,k-numbers of ditrees having dipaths of length 4. We also give bounds for ~λj,k- numbers of bipartite digraphs whose longest dipath is of length 3. Finally, we present a linear-time algorithm for determining ~λj,1-numbers of ditrees whose longest dipath is of length 3