Planar graph coloring avoiding monochromatic subgraphs: trees
نویسندگان
چکیده
We consider the problem of coloring a planar graph with the minimum number of colors so that each color class avoids one or more forbidden graphs as subgraphs. We perform a detailed study of the computational complexity of this problem. We present a complete picture for the case with a single forbidden connected (induced or noninduced) subgraph. The 2-coloring problem is NP-hard if the forbidden subgraph is a tree with at least two edges, and it is polynomially solvable in all other cases. The 3-coloring problem is NP-hard if the forbidden subgraph is a path with at least one edge, and it is polynomially solvable in all other cases. We also derive results for several forbidden sets of cycles. In particular, we prove that it is NP-complete to decide if a planar graph can be 2-colored so that no cycle of length at most 5 is monochromatic.
منابع مشابه
Chromatic Sums for Colorings Avoiding Monochromatic Subgraphs
Given graphs G and H, a vertex coloring c : V (G) → N is an H-free coloring of G if no color class contains a subgraph isomorphic to H. The H-free chromatic number of G, χ(H,G), is the minimum number of colors in an H-free coloring of G. The H-free chromatic sum of G, Σ(H,G), is the minimum value achieved by summing the vertex colors of each H-free coloring of G. We provide a general bound for ...
متن کاملMonochromatic and Heterochromatic Subgraphs in Edge-Colored Graphs - A Survey
Nowadays the term monochromatic and heterochromatic (or rainbow, multicolored) subgraphs of an edge colored graph appeared frequently in literature, and many results on this topic have been obtained. In this paper, we survey results on this subject. We classify the results into the following categories: vertex-partitions by monochromatic subgraphs, such as cycles, paths, trees; vertex partition...
متن کاملOn monochromatic subgraphs of edge-colored complete graphs
In a red-blue coloring of a nonempty graph, every edge is colored red or blue. If the resulting edge-colored graph contains a nonempty subgraph G without isolated vertices every edge of which is colored the same, then G is said to be monochromatic. For two nonempty graphs G and H without isolated vertices, the monochromatic Ramsey number mr(G,H) of G and H is the minimum integer n such that eve...
متن کاملEdge-colorings avoiding rainbow and monochromatic subgraphs
For two graphs G and H , let the mixed anti-Ramsey numbers, maxR(n; G, H), (minR(n; G, H)) be the maximum (minimum) number of colors used in an edge-coloring of a complete graph with n vertices having no monochromatic subgraph isomorphic to G and no totally multicolored (rainbow) subgraph isomorphic to H . These two numbers generalize the classical anti-Ramsey and Ramsey numbers, respectively. ...
متن کاملLecture 9: Probabilistic methods
Previously in class we proved that if we color the edges of K6 using blue or red color, then either there is a blue K3 or a red K3 as a subgraph. Here Kn is the complete graph (every pair of vertices are connected) on n vertices. We can generalize the above concept and ask, are there complete graphs for which any 2-coloring gives rise to either a blue Kk or a red Kl. It has been shown that ther...
متن کاملذخیره در منابع من
با ذخیره ی این منبع در منابع من، دسترسی به آن را برای استفاده های بعدی آسان تر کنید
عنوان ژورنال:
دوره شماره
صفحات -
تاریخ انتشار 2005