The main result of the present paper shows that the chromatic number of G² is linear in ω(G) and is at most 13ω(G), where ω(G) is the independence number of G. T., Postle, L.: Colouring graphs with sparse neighbourhoods: bounds and applications. An upper bound on the chromatic number of G² implies directly an upper bound on the frame length in the Time Division Multi-Allocation (TDMA) used for transmission scheduling in the sensor network. We show that the strong chromatic index of unit disk graphs is. Namely, if G, an UD, is an underlying graph of a wireless sensor network, the square of G, G², reflects possible conflicts in transmission between sensors in the network. Our results on the chromatic number of G2 are motivated by an application for wireless sensor networks. The square G² of G is defined as a graph on the same vertex set as G and having edges between pairs of vertices with graph distance at most two in G. In this paper, we experiment with several coloring algorithms: SmallestĪ graph G is a Unit Disk (UD) graph if there is an assignment of unit disks centered at its vertices such that there is an edge in G iff the corresponding unit disks intersect. We also point out that our novel backbone construction method is flexible such that any efficient coloring algorithm can be plugged into it. Our novel backbone generation algorithm confirms experimentally the existence of these (δ + 1) backbones in a random geometric graph, and provides an efficient topologically based centralized algorithm for determining the backbones. Hence backbone rotation in a wireless sensor network would reach all sensors (vertices) sufficiently frequently. We further show for each sample G(n, r) that the relatively few vertices not covered by all (δ + 1) backbones are covered by most of the backbones. We establish experimentally by measures of node degrees, link lengths, and interior triangular face counts that each individual backbone has most of the coverage behavior and routing convenience of the triangular ”perfect packing ” lattice. ε < 0.01), with about 30 % being fully dominating, which is consistent with the 1 4 constant approximation factor algorithm proposed recently in for the domatic partition problem in Unit Disk Graphs. For large vertex sets (n = 1600, 3200) the resulting backbones are shown to each cover typically over 99 % of the vertices of G (i.e. The backbone node sets are determined by a graph coloring algorithm employing only the topology (not the geometry) of G(n, r), and the backbone links are selected with link lengths in a narrow window between r and 2r and further to form a planar graph backbone. Using this gadget, $\frac 83$ can play the role of $\sqrt 3$ in the previous construction.For a random geometric graph G(n, r) of minimum degree δ, we introduce an efficient algorithm for selecting (δ + 1) backbones with disjoint node sets that are each independent (1 − ε) dominating sets of G. This also proves that it's NP-hard to decide whether the chromatic number of a coin graph is $3$ or $4$.Ī similar construction also shows that deciding whether a unit distance graph is $4$-colorable is also NP-complete using that there is a unit distance graph in whose only $4$-coloring two given vertices at distance $\frac 83$ must get the same color. (For this reduction, maximum degree $6$ would suffice.) We can repeat the previous construction, just if we map the vertices of $G$ far away in the plane, then the respective paths won't cross. For this, we need to use that it is NP-hard to decide whether a planar graph with maximum degree $4$ has a $3$-coloring. With a little more care, we can also show that the problem remains NP-complete if we restrict it to planar unit distance graphs. Now $G'$ is $3$-colorable if and only if $G$ is $3$-colorable if there are no additional coincidences among the vertices of $G'$ but we can pick the paths in a generic way. Add the endpoints of the segments to the vertices of $G'$, and also the two vertices at unit distance from the ends of segments of length $\sqrt 3$. Connect every two adjacent vertices with a polygonal path that consists of segments of length $\sqrt 3$ and one segment of length $1$. Map the vertices of $G$ into the plane such that no two are at unit distance.
With this gadget we can convert the decision problem of the $3$-colorability of any graph $G$ into the decision problem of the $3$-colorability of a unit distance graph $G'$as follows. They key observation is that in a $3$-coloring any two vertices $u$ and $v$ at distance $\sqrt 3$ must get the same color if the two vertices at unit distance from them are also present in the graph. Below I sketch why it is NP-hard to decide if a unit distance graph has a $3$-coloring.
0 kommentar(er)
