Cycles of Length 4 or 8 in Graphs with Diameter 2 and Minimum Degree at Least 3

Abstract

In this short note it is shown that every graph of diameter 2 and minimum degree at least 3 contains a cycle of length 4 or 8. This result contributes to the study of the Erdős-Gyárfás Conjecture by confirming it for the class of diameter-2 graphs.

Figures (8)

• Figure 1: $G'$ - Initial edge with neighbors satisfying the degree constraint.
• Figure 2: $6$-Cycle with a $v_1v_2$ Chord
• Figure 3: $v_8 \in N(v_7) \cap N(v_3)$
• Figure 4: $v_8v_9 \in E(G)$
• Figure 5: $v_{10} \in N(v_2) \cap N(v_9)$ forming an $8$-Cycle