{{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<br />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.<ref>{{MathWorld|urlname=BullGraph|title=Bull Graph}}</ref>
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,<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|volume=3|year=1987|pages=127–139|issue=1|doi=10.1007/BF01788536|s2cid=44570627}}.</ref> and a 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|volume=11|year=1995|pages=171–178|issue=2|doi=10.1007/BF01929485|s2cid=206808701}}.</ref>
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),<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|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 equal to <math>(x-2)(x-1)^3x</math>.]]
The 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 is <math>-x(x^2-x-3)(x^2+x-1)</math>.
Its Tutte polynomial is <math>x^4+x^3+x^2y</math>. {{-}}
== References == {{reflist}}
Category:Individual graphs Category:Planar graphs