Several Roman domination graph invariants on Kneser graphs

Abstract

This paper considers the following three Roman domination graph invariants on Kneser graphs: Roman domination, total Roman domination, and signed Roman domination. For Kneser graph $K_{n,k}$, we present exact values for Roman domination number $γ_{R}(K_{n,k})$ and total Roman domination number $γ_{tR}(K_{n,k})$ proving that for $n\geqslant k(k+1)$, $γ_{R}(K_{n,k}) =γ_{tR}(K_{n,k}) = 2(k+1)$. For signed Roman domination number $γ_{sR}(K_{n,k})$, the new lower and upper bounds for $K_{n,2}$ are provided: we prove that for $n\geqslant 12$, the lower bound is equal to 2, while the upper bound depends on the parity of $n$ and is equal to 3 if $n$ is odd, and equal to $5$ if $n$ is even. For graphs of smaller dimensions, exact values are found by applying exact methods from literature.

Figures (1)

• Figure 1: The Kneser graph $K_{5,2}$

Theorems & Definitions (11)

• Theorem 1
• Theorem 2
• proof
• Example 1
• Corollary 1
• proof
• Remark 1
• Remark 2
• Theorem 3
• proof