Seminar of Graph Theory - Nina Hronkovičová (11.5.2023)
Thursday 11.5.2023 at 9:50, Lecture room M/213
By: Martin Škoviera
On Siamese color graphs of small order
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 $3$.