Near-colorings: non-colorable graphs and NP-completeness

A graph G is (d_1,..,d_l)-colorable if the vertex set of G can be partitioned into subsets V_1,..,V_l such that the graph G[V_i] induced by the vertices of V_i has maximum degree at most d_i for all 1 <= i <= l. In this paper, we focus on complexity aspects of such colorings when l=2,3. More precisely, we prove that, for any fixed integers k,j,g with (k,j) distinct form (0,0) and g >= 3, either every planar graph with girth at least g is (k,j)-colorable or it is NP-complete to determine whether a planar graph with girth at least g is (k,j)-colorable. Also, for any fixed integer k, it is NP-complete to determine whether a planar graph that is either (0,0,0)-colorable or non-(k,k,1)-colorable is (0,0,0)-colorable. Additionally, we exhibit non-(3,1)-colorable planar graphs with girth 5 and non-(2,0)-colorable planar graphs with girth 7

Design of fault-tolerant on-board network

An $(n,k,r)$-network is a triple $N=(G,in,out)$ where $G=(V,E)$ is a graph and $in,out$ are integral functions defined on $V$ called input and output functions, such that for any v \inV, $in(v)+out(v)+ deg(v)\leq2r$ with $deg(v)$ the degree of $v$ in the graph $G$. The total number of inputs is in(V)=\sum_v\inVin(v)=n, and the total number of outputs is out(V)=\sum_v\inVout(v)=n+k. An $(n,k,r)$-network is valid, if for any faulty output function $out'$ (that is such that out'(v) \leqout(v) for any v \inV, and $out'(V) = n$), there are $n$ edge-disjoint paths in $G$ such that each vertex v\inV is the initial vertex of $in(v)$ paths and the terminal vertex of $out'(v)$ paths. We investigate the design problem of determining the minimum number of vertices in a valid $(n,k,r)$-network and of constructing minimum $(n,k,r)$-networks, or at least valid $(n,k,r)$-networks with a number of vertices close to the optimal value. We first show \frac3n+k2r-2+ \frac3r^2k \leq\calN(n,k,r)\leq\left\lceil\frack+22r-2\right\rceil\fracn2. We prove a better upper bound when r\geqk/2: \calN(n,k,r) \leq\fracr-2+k/2r^2-2r+k/2 n + O(1). Finally, we give the exact value of \calN(n,k,r) when $k\leq6$ and exhibit the corresponding networks

Steinberg's Conjecture and near-colorings

Let F be the family of planar graphs without cycles of length 4 and 5. Steinberg's Conjecture (1976) that says every graph of F is 3-colorable remains widely open. Focusing on a relaxation proposed by Erd˝os (1991), many studies proved the conjecture for some subfamilies of F. For example, Borodin et al. proved that every planar graph without cycles of length 4 to 7 is 3- colorable. In this note we propose to relax the problem not on the family of graphs but on the coloring by considering near-colorings. A graph G = (V,E) is said to be (i, j, k)-colorable if its vertex set can be partitioned into three sets V1, V2, V3 such that the graphs G[V1],G[V2],G[V3] induced by the sets V1, V2, V3 have maximum degree at most i, j, k respectively. Under this terminology, Steinberg's Conjecture says that every graph of F is (0, 0, 0)-colorable. A result of Xu (2008) implies that every graph of F is (1, 1, 1)-colorable. Here we prove that every graph of F is (2, 1, 0)-colorable and (4, 0, 0)-colorable

On two variations of identifying codes

Identifying codes have been introduced in 1998 to model fault-detection in multiprocessor systems. In this paper, we introduce two variations of identifying codes: weak codes and light codes. They correspond to fault-detection by successive rounds. We give exact bounds for those two definitions for the family of cycles