16,294 research outputs found
Parameterized Algorithms for Load Coloring Problem
One way to state the Load Coloring Problem (LCP) is as follows. Let
be graph and let be a 2-coloring. An
edge is called red (blue) if both end-vertices of are red (blue).
For a 2-coloring , let and be the number of red and blue edges
and let . Let be the maximum of
over all 2-colorings.
We introduce the parameterized problem -LCP of deciding whether , where is the parameter. We prove that this problem admits a kernel with
at most . Ahuja et al. (2007) proved that one can find an optimal
2-coloring on trees in polynomial time. We generalize this by showing that an
optimal 2-coloring on graphs with tree decomposition of width can be found
in time . We also show that either is a Yes-instance of -LCP
or the treewidth of is at most . Thus, -LCP can be solved in time
$O^*(4^k).
The Relaxed Game Chromatic Index of \u3cem\u3ek\u3c/em\u3e-Degenerate Graphs
The (r, d)-relaxed coloring game is a two-player game played on the vertex set of a graph G. We consider a natural analogue to this game on the edge set of G called the (r, d)-relaxed edge-coloring game. We consider this game on trees and more generally, on k-degenerate graphs. We show that if G is k-degenerate with ∆(G) = ∆, then the first player, Alice, has a winning strategy for this game with r = ∆+k−1 and d≥2k2 + 4k
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