The Bull graph
It has chromatic number 3, chromatic index 3, radius 2, diameter 3 and girth 3. It is also a block graph, a split graph, an interval graph, a claw-free graph, a 1-vertex-connected graph and a 1-edge-connected graph.
A graph is bull-free if it has no bull as an induced subgraph. The triangle-free graphs are bull-free graphs, since every bull contains a triangle. The strong perfect graph theorem was proven for bull-free graphs long before its proof for general graphs, and a polynomial time recognition algorithm for Bull-free perfect graphs is known.
Maria Chudnovsky and Shmuel Safra have studied bull-free graphs more generally, showing that any such graph must have either a large clique or a large independent set (that is, the Erdős–Hajnal conjecture holds for the bull graph), and developing a general structure theory for these graphs.
Chromatic and characteristic polynomial
The chromatic polynomial of the bull graph is . Two other graphs are chromatically equivalent to the bull graph.
Its characteristic polynomial is .
Its Tutte polynomial is .
- Weisstein, Eric W. "Bull Graph". MathWorld.
- Chvátal, V.; Sbihi, N. (1987), "Bull-free Berge graphs are perfect", Graphs and Combinatorics, 3 (1): 127–139, doi:10.1007/BF01788536.
- Reed, B.; Sbihi, N. (1995), "Recognizing bull-free perfect graphs", Graphs and Combinatorics, 11 (2): 171–178, doi:10.1007/BF01929485.
- Chudnovsky, M.; Safra, S. (2008), "The Erdős–Hajnal conjecture for bull-free graphs", Journal of Combinatorial Theory, Series B, 98 (6): 1301–1310, doi:10.1016/j.jctb.2008.02.005.
- Chudnovsky, M. (2008), The structure of bull-free graphs. I. Three-edge paths with centers and anticenters (PDF).
- Chudnovsky, M. (2008), The structure of bull-free graphs. II. Elementary trigraphs (PDF).
- Chudnovsky, M. (2008), The structure of bull-free graphs. III. Global structure (PDF).