Bull graph

In the mathematical field of graph theory, the bull graph is a planar undirected graph with 5 vertices and 5 edges, in the form of a triangle with two disjoint pendant edges.[1]

Bull graph
The bull graph
Vertices5
Edges5
Radius2
Diameter3
Girth3
Automorphisms2 (Z/2Z)
Chromatic number3
Chromatic index3
PropertiesPlanar
Unit distance
Table of graphs and parameters

It has chromatic number 3, chromatic index 3, radius 2, diameter 3 and girth 3. It is also a self-complementary graph, a block graph, a split graph, an interval graph, a claw-free graph, a 1-vertex-connected graph and a 1-edge-connected graph.

Bull-free graphs

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,[2] and a polynomial time recognition algorithm for Bull-free perfect graphs is known.[3]

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),[4] and developing a general structure theory for these graphs.[5][6][7]

Chromatic and characteristic polynomial
The three graphs with a chromatic polynomial equal to .

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 .

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References
  1. Weisstein, Eric W. "Bull Graph". MathWorld.
  2. Chvátal, V.; Sbihi, N. (1987), "Bull-free Berge graphs are perfect", Graphs and Combinatorics, 3 (1): 127–139, doi:10.1007/BF01788536.
  3. Reed, B.; Sbihi, N. (1995), "Recognizing bull-free perfect graphs", Graphs and Combinatorics, 11 (2): 171–178, doi:10.1007/BF01929485.
  4. 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, archived from the original on 2010-06-25, retrieved 2009-06-30.
  5. Chudnovsky, M. (2008), The structure of bull-free graphs. I. Three-edge paths with centers and anticenters (PDF).
  6. Chudnovsky, M. (2008), The structure of bull-free graphs. II. Elementary trigraphs (PDF).
  7. Chudnovsky, M. (2008), The structure of bull-free graphs. III. Global structure (PDF).
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