2 research outputs found
Minimum Sum Edge Colorings of Multicycles
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 {\em 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
On the cost chromatic number of outerplanar, planar, and line graphs
We consider vertex colorings of graphs in which each color has an associated cost which is incurred each time the color is assigned to a vertex. The cost of the coloring is the sum of the costs incurred at each vertex. The cost chromatic number of a graph with respect to a cost set is the minimum number of colors necessary to produce a minimum cost coloring of the graph. We show that the cost chromatic number of maximal outerplanar and maximal planar graphs can be arbitrarily large and construct several infinite classes of counterexamples to a conjecture of Harary and Plantholt on the cost chromatic number of line graphs