On minimally b-imperfect graphs
نویسندگان
چکیده
A b-coloring is a coloring of the vertices of a graph such that each color class contains a vertex that has a neighbor in all other color classes. The b-chromatic number of a graph G is the largest integer k such that G admits a b-coloring with k colors. A graph is b-perfect if the b-chromatic number is equal to the chromatic number for every induced subgraph H of G. A graph is minimally b-imperfect if it is not b-perfect and every proper induced subgraph is b-perfect. We give a list F of minimally b-imperfect graphs, conjecture that a graph is b-perfect if and only if it does not contain a graph from this list as an induced subgraph, and prove this conjecture for several classes of graphs, namely diamond-free graphs, and graphs with chromatic number at most three.
منابع مشابه
Critical Edges in Perfect Graphs and Some Polyhedral Consequences
An edge e of a perfect graph G is called critical if G ? e is imperfect. For certain graphs G ? e of this type, we determine all minimally imperfect subgraphs. We use this knowledge to describe inequalities inducing facets of the stable set polytope associated with G ? e.
متن کاملThe Strong Perfect Graph Conjecture
A graph is perfect if, in all its induced subgraphs, the size of a largest clique is equal to the chromatic number. Examples of perfect graphs include bipartite graphs, line graphs of bipartite graphs and the complements of such graphs. These four classes of perfect graphs will be called basic. In 1960, Berge formulated two conjectures about perfect graphs, one stronger than the other. The weak...
متن کاملThe Guessing Number of Undirected Graphs
Riis [Electron. J. Combin., 14(1):R44, 2007] introduced a guessing game for graphs which is equivalent to finding protocols for network coding. In this paper we prove upper and lower bounds for the winning probability of the guessing game on undirected graphs. We find optimal bounds for perfect graphs and minimally imperfect graphs, and present a conjecture relating the exact value for all grap...
متن کاملOn a certain class of nonideal clutters
In this paper we define the class of near-ideal clutters following a similar concept due to Shepherd [Near perfect matrices, Math. Programming 64 (1994) 295–323] for near-perfect graphs. We prove that near-ideal clutters give a polyhedral characterization for minimally nonideal clutters as near-perfect graphs did for minimally imperfect graphs. We characterize near-ideal blockers of graphs as b...
متن کاملPerfect Graphs, Partitionable Graphs and Cutsets
A graph G is perfect if, for all induced subgraphs of G, the size of a largest clique is equal to the chromatic number. A graph is minimally imperfect if it is not perfect but all its proper induced subgraphs are. A hole is a chordless cycle of length at least four. The strong perfect graph conjecture of Berge [1] states that G is minimally imperfect if and only if G or its complement is an odd...
متن کاملذخیره در منابع من
با ذخیره ی این منبع در منابع من، دسترسی به آن را برای استفاده های بعدی آسان تر کنید
برای دانلود متن کامل این مقاله و بیش از 32 میلیون مقاله دیگر ابتدا ثبت نام کنید
ثبت ناماگر عضو سایت هستید لطفا وارد حساب کاربری خود شوید
ورودعنوان ژورنال:
- Discrete Applied Mathematics
دوره 157 شماره
صفحات -
تاریخ انتشار 2009