Labeling planar graphs with a condition at distance two

An $L(2,1)$-labeling of a graph is a mapping $c:V(G) \to \{0,\ldots,K\}$ such that the labels assigned to neighboring vertices differ by at least $2$ and the labels of vertices at distance two are different. Griggs and Yeh [SIAM J. Discrete Math. 5 (1992), 586―595] conjectured that every graph $G$ with maximum degree $\Delta$ has an $L(2,1)$-labeling with $K \leq \Delta^2$. We verify the conjecture for planar graphs with maximum degree $\Delta \neq 3$

k-L(2,1)-Labelling for Planar Graphs is NP-Complete for k >= 4

A mapping from the vertex set of a graph G = (V,E) into an interval of integers {0,...,k} is an L(2,1)-labelling of G of span k if any two adjacent vertices are mapped onto integers that are at least 2 apart, and every two vertices with a common neighbour are mapped onto distinct integers. It is known that for any fixed k >= 4, deciding the existence of such a labelling is an NP-complete problem while it is polynomial for k = 8, it remains NP-complete when restricted to planar graphs. In this paper, we show that it remains NP-complete for any k >= 4 by reduction from Planar Cubic Two-Colourable Perfect Matching. Schaefer stated without proof that Planar Cubic Two-Colourable Perfect Matching is NP-complete. In this paper we give a proof of this.Comment: 16 pages, includes figures generated using PSTricks. To appear in Discrete Applied Mathematics. Some very minor corrections incorporate

L(2,1)-labelling of graphs

International audienceAn $L(2,1)$-labelling of a graph is a function $f$ from the vertex set to the positive integers such that $|f(x)-f(y)|\geq 2$ if $dist(x,y)=1$ and $|f(x)-f(y)|\geq 1$ if $dist(x,y)=2$, where $dist(u,v)$ is the distance between the two vertices~$u$ and~$v$ in the graph $G$. The \emph{span} of an $L(2,1)$-labelling $f$ is the difference between the largest and the smallest labels used by $f$ plus $1$. In 1992, Griggs and Yeh conjectured that every graph with maximum degree $\Delta\geq 2$ has an $L(2,1)$-labelling with span at most $\Delta^2+1$. We settle this conjecture for $\Delta$ sufficiently large.Un $L(2,1)$-étiquettage d'un graphe est une fonction $f$ de l'ensemble des sommets vers les entiers positifs telle que $|f(x)-f(y)|\geq 2$ si $dist(x,y)=1$ et $|f(x)-f(y)|\geq 1$ si $dist(x,y)=2$, où $dist(u,v)$ est la distance entre les sommets~$u$ et~$v$ dans le graphe $G$. Le \emph{span} d'un $L(2,1)$-étiquettage $f$ est la différence entre la plus grande et la plus petite étiquette utilisée par $f$ plus $1$. En 1992, Griggs et Yeh ont conjecturé que tout graphe de degré maximum $\Delta\geq 2$ a un $L(2,1)$-étiquettage de span au plus $\Delta^2+1$. Nous confirmons cette conjecture pour $\Delta$ suffisamment grand

L(2,1)-labelling of graphs

International audienceAn $L(2,1)$-labelling of a graph is a function $f$ from the vertex set to the positive integers such that $|f(x)-f(y)|\geq 2$ if $dist(x,y)=1$ and $|f(x)-f(y)|\geq 1$ if $dist(x,y)=2$, where $dist(u,v)$ is the distance between the two vertices~$u$ and~$v$ in the graph $G$. The \emph{span} of an $L(2,1)$-labelling $f$ is the difference between the largest and the smallest labels used by $f$ plus $1$. In 1992, Griggs and Yeh conjectured that every graph with maximum degree $\Delta\geq 2$ has an $L(2,1)$-labelling with span at most $\Delta^2+1$. We settle this conjecture for $\Delta$ sufficiently large.Un $L(2,1)$-étiquettage d'un graphe est une fonction $f$ de l'ensemble des sommets vers les entiers positifs telle que $|f(x)-f(y)|\geq 2$ si $dist(x,y)=1$ et $|f(x)-f(y)|\geq 1$ si $dist(x,y)=2$, où $dist(u,v)$ est la distance entre les sommets~$u$ et~$v$ dans le graphe $G$. Le \emph{span} d'un $L(2,1)$-étiquettage $f$ est la différence entre la plus grande et la plus petite étiquette utilisée par $f$ plus $1$. En 1992, Griggs et Yeh ont conjecturé que tout graphe de degré maximum $\Delta\geq 2$ a un $L(2,1)$-étiquettage de span au plus $\Delta^2+1$. Nous confirmons cette conjecture pour $\Delta$ suffisamment grand

Griggs and Yeh's Conjecture and L(p,1)-labelings

International audienceAn L(p,1)-labeling of a graph is a function f from the vertex set to the positive integers such that |f(x) − f(y)| ≥ p if dist(x, y) = 1 and |f(x) − f(y)| ≥ 1 if dist(x, y) = 2, where dist(x,y) is the distance between the two vertices x and y in the graph. The span of an L(p,1)- labeling f is the difference between the largest and the smallest labels used by f. In 1992, Griggs and Yeh conjectured that every graph with maximum degree Δ ≥ 2 has an L(2, 1)-labeling with span at most Δ^2. We settle this conjecture for Δ sufficiently large. More generally, we show that for any positive integer p there exists a constant Δ_p such that every graph with maximum degree Δ ≥ Δ_p has an L(p,1)-labeling with span at most Δ^2. This yields that for each positive integer p, there is an integer C_p such that every graph with maximum degree Δ has an L(p,1)-labeling with span at most Δ^2 + C_p

Labeling planar graphs with a condition at distance two

AbstractAn L(2,1)-labeling of a graph is a mapping c:V(G)→{0,…,K} such that the labels assigned to neighboring vertices differ by at least 2 and the labels of vertices at distance two are different. The smallest K for which an L(2,1)-labeling of a graph G exists is denoted by λ2,1(G). Griggs and Yeh [J.R. Griggs, R.K. Yeh, Labeling graphs with a condition at distance 2, SIAM J. Discrete Math. 5 (1992) 586–595] conjectured that λ2,1(G)≤Δ2 for every graph G with maximum degree Δ≥2. We prove the conjecture for planar graphs with maximum degree Δ≠3. All our results also generalize to the list-coloring setting

Labeling planar graphs with a condition at distance two

An L(2, 1)-labeling of a graph is a mapping c: V (G) →{0,...,K} such that the labels assigned to neighboring vertices differ by at least 2 and the labels of vertices at distance two are different. The smallest K for which an L(2, 1)-labeling of a graph G exists is denoted by λ2,1(G). Griggs and Yeh [SIAM J. Discrete Math. 5 (1992), 586–595] conjectured that λ2,1(G) ≤ ∆ 2 for every graph G withmaximumdegree∆.Weprove the conjecture for planar graphs with maximum degree ∆ = 3. Allour results also generalize to the list-coloring setting