An introduction to chromatic sums
CSC '89 Proceedings of the 17th conference on ACM Annual Computer Science Conference
On chromatic sums and distributed resource allocation
Information and Computation
Minimum color sum of bipartite graphs
Journal of Algorithms
Routing with Minimum Wire Length in the Dogleg-Free Manhattan Model is $\cal NP$-Complete
SIAM Journal on Computing
Minimal coloring and strength of graphs
Discrete Mathematics
Edge-chromatic sum of trees and bounded cyclicity graphs
Information Processing Letters
Approximation Results for the Optimum Cost Partition Problem
ICALP '97 Proceedings of the 24th International Colloquium on Automata, Languages and Programming
A 27/26-Approximation Algorithm for the Chromatic Sum Coloring of Bipartite Graphs
APPROX '02 Proceedings of the 5th International Workshop on Approximation Algorithms for Combinatorial Optimization
Discrete Applied Mathematics
Approximating Min Sum Set Cover
Algorithmica
Graphs and Hypergraphs
The complexity of chromatic strength and chromatic edge strength
Computational Complexity
Proceedings of the nineteenth annual ACM-SIAM symposium on Discrete algorithms
Minimum sum set coloring of trees and line graphs of trees
Discrete Applied Mathematics
On sum edge-coloring of regular, bipartite and split graphs
Discrete Applied Mathematics
| Hi-index | 0.05 |
In the minimum sum edge coloring problem, we aim to assign natural numbers to edges of a graph, so that adjacent edges receive different numbers, and the sum of the numbers assigned to the edges is minimum. The chromatic edge strength of a graph is the minimum number of colors required in a minimum sum edge coloring of this graph. We study the case of multicycles, defined as cycles with parallel edges, and give a closed-form expression for the chromatic edge strength of a multicycle, thereby extending a theorem due to Berge. It is shown that the minimum sum can be achieved with a number of colors equal to the chromatic index. We also propose simple algorithms for finding a minimum sum edge coloring of a multicycle. Finally, these results are generalized to a large family of minimum cost coloring problems.