Symmetric Perfect $2$-colorings on $J(10,3)$

Abstract

We study perfect $2$-coloring of the Johnson graphs $J(n,3)$ associated with the third largest eigenvalue and symmetric quotient matrix, which exists only when $n \in \{6, 10\}$. We survey the known constructions in the case $n=6$, give a new construction for the two known perfect $2$-colorings in the case $n=10$, and prove that these are the only possible ones.

Figures (2)

• Figure 1: Induced subgraphs of symmetric perfect $2$-colorings of $J(6,3)$
• Figure 2: Visualisation of the orbits of $\mathop{\mathrm{Aut}}\nolimits(\mathcal{G})$ acting on $J(10,3)$

Theorems & Definitions (19)

• Lemma 1: 3
• Lemma 2
• Lemma 3
• Lemma 4
• Lemma 5
• Lemma 6
• Lemma 7
• proof
• Lemma 8
• proof