Graph unique-maximum and conflict-free colorings

Panagiotis Cheilaris, Géza Tóth
2011 Journal of Discrete Algorithms  
We investigate the relationship between two kinds of vertex colorings of graphs: uniquemaximum colorings and conflict-free colorings. In a unique-maximum coloring, the colors are ordered, and in every path of the graph the maximum color appears only once. In a conflict-free coloring, in every path of the graph there is a color that appears only once. We also study computational complexity aspects of conflict-free colorings and prove a completeness result. Finally, we improve lower bounds for
more » ... se chromatic numbers of the grid graph. Definition 3. A graph X is a minor of Y , denoted as X Y , if X can be obtained from Y by a sequence of the following three operations: vertex deletion, edge deletion, and edge contraction. Edge contraction is the process of merging both endpoints of an edge into a new vertex, which is connected to all vertices adjacent to the two endpoints. Given a uniquemaximum coloring C of Y , we get the induced coloring of X as follows. Take a sequence of vertex deletions, edge deletions, and edge contractions so that we obtain X from Y . For the vertex and edge deletion operations, just keep the colors of the remaining vertices. For the edge contraction operation, say along edge xy, which gives rise to the new vertex v xy , set C (v xy ) = max(C (x), C (y)), and keep the colors of all other vertices. Proposition 4. (See [3].) If X Y , and C is a unique-maximum coloring of Y , then the induced coloring C is a unique-maximum coloring of X . Consequently, χ um (X) χ um (Y ). The (traditional) chromatic number of a graph is denoted by χ (G) and is the smallest number of colors in a vertex coloring for which adjacent vertices are assigned different colors. A simple relation between the chromatic numbers we have defined so far is the following. Fact 5. For every graph G, χ (G) χ cf (G) χ um (G).
doi:10.1016/j.jda.2011.03.005 fatcat:cxnfwcqytfgejfbw3cao75jxye