• # Seminár z teórie grafov - Nina Hronkovičová (11.5.2023)

## vo štvrtok 11.5.2023 o 9:50 hod. v miestnosti M/213

09. 05. 2023 20.27 hod.

Prednášajúci: Nina Hronkovičová

Názov: On Siamese color graphs of small order

Termín: 11.5.2023, 9:50 hod., M 213

Abstrakt:
A generalized quadrangle of order \$q\$ is an incidence structure such that on every line there are exactly \$q+1\$ points, at every point there intersect exactly \$q+1\$ lines, and no two distinct points lay on the two distinct lines. A spread in a generalized quadrangle is a set of lines which partition the point set. Generalized quadrangles play important role in various parts of mathematics, for example, their incidence graphs are Moore graphs of girth \$8\$. In 2003 Kharabani and Thorabi showed that for any prime power \$q\$ there exists a system of \$q+1\$ generalized quadrangles of order \$q\$ on the same set of points sharing a common spread such each pair of points lies on a line in at least one of the generalized quadrangles. They called such system a geometric Siamase color graph of order \$q\$. Klin, Reichard and Woldar showed that lines of generalized quadrangles in a geometric Siamese color graph of order \$q\$ form a Steiner system with parameters \$(2,q+1,q^3+q^2+q+1)\$ and they used this observation to classify geometric Siamese color graphs of order \$2\$. Later they found hundreds of geometric Siamese color graph order \$3\$. Using algebraic properties of generalized quadrangles with a spread derived by Brouwer in \$1984\$ we completely classify geometric Siamese color graphs of order \$2\$ and \$3\$.

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