Subcoloring

Abstract: We consider a weighted version of the subcoloring problem that we call the hypocoloring problem: given a weighted graph G=(V,E;w) where w(v)≥ 0, the goal consists in finding a partition ${\cal S}=(S_1,\ldots,S_k)$ of the node set of G into hypostable sets and minimizing ∑$_{i=1}^{k}$w(Si) where an hypostable S is a subset of nodes which generates a collection of node disjoint cliques K. The weight of S is defined as max {∑v∈Kw(v)| K∈S}. Properties of hypocolorings are stated; complexity and approximability results are presented in some graph classes. The associated decision problem is shown to be NP-complete for bipartite graphs and triangle-free planar graphs with maximum degree 3. Polynomial algorithms are given for graphs with maximum degree 2 and for trees with maximum degree Δ.

Abstract: For any $$k \ge 2$$ki?ź2, deciding whether the linear arboricity, star arboricity, caterpillar arboricity, and spider arboricity, respectively, of a bipartite graph are at most k are all NP-complete.

Abstract: A partition of the vertex set V(G) of a graph G into $$V(G)=V_1\cup V_2\cup \cdots \cup V_k$$V(G)=V1źV2źźźVk is called a k-strong subcoloring if $$d(x,y) e 2$$d(x,y)ź2 in G for every $$x,y\in V_i$$x,yźVi, $$1\le i \le k$$1≤i≤k where d(x, y) denotes the length of a shortest x-y path in G. The strong subchromatic number is defined as $$\chi _{sc}(G)=\text {min}\{ k:G \text { admits a }k$$źsc(G)=min{k:Gadmits ak-$$\text {strong subcoloring}\}$$strong subcoloring}. In this paper, we explore the complexity status of the StrongSubcoloring problem: for a given graph G and a positive integer k, StrongSubcoloring is to decide whether G admits a k-strong subcoloring. We prove that StrongSubcoloring is NP-complete for subcubic bipartite graphs and the problem is polynomial time solvable for trees. In addition, we prove the following dichotomy results: (i) for the class of $$K_{1,r}$$K1,r-free split graphs, StrongSubcoloring is in P when $$r\le 3$$r≤3 and NP-complete when $$r>3$$r>3 and (ii) for the class of H-free graphs, StrongSubcoloring is polynomial time solvable only if H is an induced subgraph of $$P_4$$P4; otherwise the problem is NP-complete. Next, we consider a lower bound on the strong subchromatic number. A strong set is a set S of vertices of a graph G such that for every $$x,y\in S$$x,yźS, $$d(x,y)= 2$$d(x,y)=2 in G and the cardinality of a maximum strong set in G is denoted by $$\alpha _{s}(G)$$źs(G). Clearly, $$\alpha _{s}(G)\le \chi _{sc}(G)$$źs(G)≤źsc(G). We consider the complexity status of the StrongSet problem: given a graph G and a positive integer k, StrongSet asks whether G contains a strong set of cardinality k. We prove that StrongSet is NP-complete for (i) bipartite graphs and (ii) $$K_{1,4}$$K1,4-free split graphs, and it is polynomial time solvable for (i) trees and (ii) $$P_4$$P4-free graphs.

Abstract: We consider a weighted version of the subcoloring problem that we call the hypocoloring problem: given a weighted graph G = (V,E;w) where w(v) > 0, the goal consists in finding a partition S = (S 1 ,..., S k ) of the node set of G into hypostable sets and minimizing Σ k i=1 w(S i ) where an hypostable S is a subset of nodes which generates a collection of node disjoint cliques K. The weight of S is defined as max{Σ v ∈ K w(v)| K E S}. Properties of hypocolorings are stated; complexity and approximability results are presented in some graph classes. The associated decision problem is shown to be NP-complete for bipartite graphs and triangle-free planar graphs with maximum degree 3. Polynomial algorithms are given for graphs with maximum degree 2 and for trees with maximum degree Δ.