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EXTREMAL GRAPHS FOR DEGREE SUMS AND DOMINATING CYCLES
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- Journal:
- Bulletin of the Australian Mathematical Society , First View
- Published online by Cambridge University Press:
- 13 September 2024, pp. 1-7
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A cycle C of a graph G is dominating if
$V(C)$ is a dominating set and 
$V(G)\backslash V(C)$ is an independent set. Wu et al. [‘Degree sums and dominating cycles’, Discrete Mathematics 344 (2021), Article no. 112224] proved that every longest cycle of a k-connected graph G on 
$n\geq 3$ vertices with 
$k\geq 2$ is dominating if the degree sum is more than 
$(k+1)(n+1)/3$ for any 
$k+1$ pairwise nonadjacent vertices. They also showed that this bound is sharp. In this paper, we show that the extremal graphs G for this condition satisfy 
$(n-2)/3K_1\vee (n+1)/3K_2 \subseteq G \subseteq K_{(n-2)/3}\vee (n+1)/3K_2$ or 
$2K_1\vee 3K_{(n-2)/3}\subseteq G \subseteq K_2\vee 3K_{(n-2)/3}.$
