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- Brinkmann_graph abstract "In the mathematical field of graph theory, the Brinkmann graph is a 4-regular graph with 21 vertices and 42 edges discovered by Gunnar Brinkmann in 1992. It was first published by Brinkmann and Meringer in 1997.It has chromatic number 4, chromatic index 5, radius 3, diameter 3 and girth 5. It is also a 3-vertex-connected graph and a 3-edge-connected graph.By Brooks’ theorem, every k-regular graph (except for odd cycles and cliques) has chromatic number at most k. It was also known since 1959 that, for every k and l there exist k-chromatic graphs with girth l. In connection with these two results and several examples including the Chvátal graph, Branko Grünbaum conjectured in 1970 that for every k and l there exist k-chromatic k-regular graphs with girth l. The Chvátal graph solves the case k = l = 4 of this conjecture and the Brinkmann graph solves the case k = 4, l = 5. Grünbaum's conjecture was disproved for sufficiently large k by Johannsen, who showed that the chromatic number of a triangle-free graph is O(Δ/log Δ) where Δ is the maximum vertex degree and the O introduces big O notation. However, despite this disproof, it remains of interest to find examples and only very few are known.The chromatic polynomial of the Brinkmann graph is x21 - 42x20 + 861x19 - 11480x18 + 111881x17 - 848708x16 + 5207711x15 - 26500254x14 + 113675219x13 - 415278052x12 + 1299042255x11 - 3483798283x10 + 7987607279x9 - 15547364853x8 + 25384350310x7 - 34133692383x6 + 36783818141x5 - 30480167403x4 + 18168142566x3 - 6896700738x2 + 1242405972x (sequence A159192 in OEIS).".
- Brinkmann_graph thumbnail Brinkmann_graph_LS.svg?width=300.
- Brinkmann_graph wikiPageID "24177385".
- Brinkmann_graph wikiPageRevisionID "550110424".
- Brinkmann_graph automorphisms "14".
- Brinkmann_graph chromaticIndex "5".
- Brinkmann_graph chromaticNumber "4".
- Brinkmann_graph diameter "3".
- Brinkmann_graph edges "42".
- Brinkmann_graph girth "5".
- Brinkmann_graph hasPhotoCollection Brinkmann_graph.
- Brinkmann_graph imageCaption "The Brinkmann graph".
- Brinkmann_graph name "Brinkmann graph".
- Brinkmann_graph namesake "Gunnar Brinkmann".
- Brinkmann_graph properties Eulerian_path.
- Brinkmann_graph properties Hamiltonian_path.
- Brinkmann_graph radius "3".
- Brinkmann_graph vertices "21".
- Brinkmann_graph subject Category:4-chromatic_graphs.
- Brinkmann_graph subject Category:Individual_graphs.
- Brinkmann_graph subject Category:Regular_graphs.
- Brinkmann_graph type Abstraction100002137.
- Brinkmann_graph type Communication100033020.
- Brinkmann_graph type Graph107000195.
- Brinkmann_graph type IndividualGraphs.
- Brinkmann_graph type RegularGraphs.
- Brinkmann_graph type VisualCommunication106873252.
- Brinkmann_graph comment "In the mathematical field of graph theory, the Brinkmann graph is a 4-regular graph with 21 vertices and 42 edges discovered by Gunnar Brinkmann in 1992. It was first published by Brinkmann and Meringer in 1997.It has chromatic number 4, chromatic index 5, radius 3, diameter 3 and girth 5. It is also a 3-vertex-connected graph and a 3-edge-connected graph.By Brooks’ theorem, every k-regular graph (except for odd cycles and cliques) has chromatic number at most k.".
- Brinkmann_graph label "Brinkmann graph".
- Brinkmann_graph label "Graphe de Brinkmann".
- Brinkmann_graph sameAs Graphe_de_Brinkmann.
- Brinkmann_graph sameAs m.07kjpkw.
- Brinkmann_graph sameAs Q3115474.
- Brinkmann_graph sameAs Q3115474.
- Brinkmann_graph sameAs Brinkmann_graph.
- Brinkmann_graph wasDerivedFrom Brinkmann_graph?oldid=550110424.
- Brinkmann_graph depiction Brinkmann_graph_LS.svg.
- Brinkmann_graph isPrimaryTopicOf Brinkmann_graph.
