# Bull graph

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{{Infobox graph
 | name = Bull graph
 | image = 170px
 | image_caption = The bull graph
 | namesake =
 | vertices = 5
 | edges = 5
 | automorphisms = 2 ('''Z'''/2'''Z''')
 | diameter = 3
 | girth = 3
 | radius = 2
 | chromatic_number = 3
 | chromatic_index = 3
 | properties = [Planar](/source/planar_graph)<br />[Unit distance](/source/unit_distance_graph)
}}

In the [mathematical](/source/mathematics) field of [graph theory](/source/graph_theory), the '''bull graph''' is a [planar](/source/planar_graph) [undirected graph](/source/undirected_graph) with 5 vertices and 5 edges, in the form of a triangle with two disjoint pendant edges.<ref>{{MathWorld|urlname=BullGraph|title=Bull Graph}}</ref>

It has [chromatic number](/source/chromatic_number) 3, [chromatic index](/source/chromatic_index) 3, radius 2, diameter 3 and [girth](/source/girth_(graph_theory)) 3. It is also a [self-complementary graph](/source/self-complementary_graph), a [block graph](/source/block_graph), a [split graph](/source/split_graph), an [interval graph](/source/interval_graph), a [claw-free graph](/source/claw-free_graph), a 1-[vertex-connected graph](/source/k-vertex-connected_graph) and a 1-[edge-connected graph](/source/k-edge-connected_graph).

== Bull-free graphs ==
A graph is bull-free if it has no bull as an [induced subgraph](/source/induced_subgraph). The [triangle-free graph](/source/triangle-free_graph)s are bull-free graphs, since every bull contains a triangle. The [strong perfect graph theorem](/source/strong_perfect_graph_theorem) was proven for bull-free graphs long before its proof for general graphs,<ref>{{citation|last1=Chvátal|first1=V.|author1-link=Václav Chvátal|last2=Sbihi|first2=N.|author2-link=Najiba Sbihi|title=Bull-free Berge graphs are perfect|journal=[Graphs and Combinatorics](/source/Graphs_and_Combinatorics)|volume=3|year=1987|pages=127–139|issue=1|doi=10.1007/BF01788536|s2cid=44570627}}.</ref> and a [polynomial time](/source/polynomial_time) recognition algorithm for Bull-free perfect graphs is known.<ref>{{citation|last1=Reed|first1=B.|author1-link=Bruce Reed (mathematician)|last2=Sbihi|first2=N.|author2-link=Najiba Sbihi|title=Recognizing bull-free perfect graphs|journal=[Graphs and Combinatorics](/source/Graphs_and_Combinatorics)|volume=11|year=1995|pages=171–178|issue=2|doi=10.1007/BF01929485|s2cid=206808701}}.</ref>

[Maria Chudnovsky](/source/Maria_Chudnovsky) and [Shmuel Safra](/source/Shmuel_Safra) have studied bull-free graphs more generally, showing that any such graph must have either a large [clique](/source/clique_(graph_theory)) or a large [independent set](/source/independent_set_(graph_theory)) (that is, the [Erdős–Hajnal conjecture](/source/Erd%C5%91s%E2%80%93Hajnal_conjecture) holds for the bull graph),<ref>{{citation|last1=Chudnovsky|first1=M.|author1-link=Maria Chudnovsky|last2=Safra|first2=S.|title=The Erdős–Hajnal conjecture for bull-free graphs|journal=[Journal of Combinatorial Theory](/source/Journal_of_Combinatorial_Theory)|series=Series B|volume=98|issue=6|year=2008|pages=1301–1310|doi=10.1016/j.jctb.2008.02.005|doi-access=free|citeseerx=10.1.1.606.3091}}.</ref> and developing a general structure theory for these graphs.<ref>{{citation|last=Chudnovsky|first=M.|authorlink=Maria Chudnovsky|year=2008|title=The structure of bull-free graphs. I. Three-edge paths with centers and anticenters|url=http://www.columbia.edu/~mc2775/bulls1.pdf}}.</ref><ref>{{citation|last=Chudnovsky|first=M.|authorlink=Maria Chudnovsky|year=2008|title=The structure of bull-free graphs. II. Elementary trigraphs|url=http://www.columbia.edu/~mc2775/bulls2.pdf}}.</ref><ref>{{citation|last=Chudnovsky|first=M.|authorlink=Maria Chudnovsky|year=2008|title=The structure of bull-free graphs. III. Global structure|url=http://www.columbia.edu/~mc2775/bulls3.pdf}}.</ref>

== Chromatic and characteristic polynomial ==
[[File:Chromatically equivalent graphs.svg|thumb|200px|The three graphs with a [chromatic polynomial](/source/chromatic_polynomial) equal to <math>(x-2)(x-1)^3x</math>.]]

The [chromatic polynomial](/source/chromatic_polynomial) of the bull graph is <math>(x-2)(x-1)^3x</math>. Two other graphs are chromatically equivalent to the bull graph.

Its [characteristic polynomial](/source/characteristic_polynomial) is <math>-x(x^2-x-3)(x^2+x-1)</math>.

Its [Tutte polynomial](/source/Tutte_polynomial) is <math>x^4+x^3+x^2y</math>.
{{-}}

== References ==
{{reflist}}

Category:Individual graphs
Category:Planar graphs

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Adapted from the Wikipedia article [Bull graph](https://en.wikipedia.org/wiki/Bull_graph) by Wikipedia contributors ([contributor history](https://en.wikipedia.org/wiki/Bull_graph?action=history)). Available under [Creative Commons Attribution-ShareAlike 4.0 International](https://creativecommons.org/licenses/by-sa/4.0/). Changes may have been made.