# Bull graph

Bull graph
The Bull graph
Vertices 5
Edges 5
Diameter 3
Girth 3
Automorphisms 2 (Z/2Z)
Chromatic number 3
Chromatic index 3
Properties Planar
Unit distance

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]

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.

## 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 ${\displaystyle (x-2)(x-1)^{3}x}$.

The chromatic polynomial of the bull graph is ${\displaystyle (x-2)(x-1)^{3}x}$. Two other graphs are chromatically equivalent to the bull graph.

Its characteristic polynomial is ${\displaystyle -x(x^{2}-x-3)(x^{2}+x-1)}$.

Its Tutte polynomial is ${\displaystyle x^{4}+x^{3}+x^{2}y}$.

## References

1. ^
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.
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