Robert Janczewski

A 27/26-Approximation Algorithm for the Chromatic Sum Coloring of Bipartite Graphs

We consider the CHROMATIC SUM PROBLEM on bipartite graphs which appears to be much harder than the classical CHROMATIC NUMBER PROBLEM. We prove that the CHROMATIC SUM PROBLEM is NP-complete on planar bipartite graphs with ? ? 5, but polynomial on bipartite graphs with ? ? 3, for which we construct an O(n2)-time algorithm. Hence, we tighten the borderline of intractability for this problem on bipartite graphs with bounded degree, namely: the case ? = 3 is easy, ? = 5 is hard. Moreover, we construct a 27/26-approximation algorithm for this problem thus improving the best known approximation ratio of 10/9.

The complexity of the L(p,q) -labeling problem for bipartite planar graphs of small degree

Given a simple graph G, by an L(p,q)-labeling of G we mean a function c that assigns nonnegative integers to its vertices in such a way that if two vertices u, v are adjacent then |c(u)-c(v)|>=p, and if they are at distance 2 then |c(u)-c(v)|>=q. The L(p,q)-labeling problem can be defined as follows: given a graph G and integer t, determine whether there exists an L(p,q)-labeling c of G such that c(V)@?{0,1,...,t}. In the paper we show that the problem is NP-complete even when restricted to bipartite planar graphs of small maximum degree and for relatively small values of t. More precisely, we prove that: (1)if p 3q then the problem is NP-complete for bipartite planar graphs of maximum degree @D@?4 and t=p+5q. In particular, these results imply that the L(2,1)-labeling problem in planar graphs is NP-complete for t=4, and that the L(p,q)-labeling problem in graphs of maximum degree @D@?4 is NP-complete for all values of p and q, thus answering two well-known open questions.

Sum Coloring of Bipartite Graphs with Bounded Degree

Abstract We consider the Chromatic Sum Problem on bipartite graphs which appears to be much harder than the classical Chromatic Number Problem. We prove that the Chromatic Sum Problem is NP-complete on planar bipartite graphs with

Note: Greedy T-colorings of graphs

\Delta \leq 5

The smallest hard-to-color graph for algorithm DSATUR

, but polynomial on bipartite graphs with

The complexity of the T -coloring problem for graphs with small degree

\Delta \leq 3

A polynomial algorithm for finding T -span of generalized cacti

, for which we construct an

Divisibility and T -span of graphs

O(n^{2})

Interval wavelength assignment in all-optical star networks

-time algorithm. Hence, we tighten the borderline of intractability for this problem on bipartite graphs with bounded degree, namely: the case

The computational complexity of the backbone coloring problem for bounded-degree graphs with connected backbones

\Delta =3