Note: Some results on Vizing's conjecture and related problems
Discrete Applied Mathematics
FCT'13 Proceedings of the 19th international conference on Fundamentals of Computation Theory
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Let $P_2(G)$, $\gf(G)$, and $\gm(G)$ be the 2-packing number, fractional domination number, and domination number, respectively, of a graph $G$. Domke, Hedetniemi, and Laskar [Congress. Numer., 66 (1989), pp. 227--238] showed that $P_2(G)\le\gf(G)\le\gm(G)$. Examples are given with $P_2(G)Cartesian product and strong direct product, respectively, of graphs $G$ and $H$. For all $G$ and $H$, it is shown that $P_2(G)P_2(H)\le P_2(G\cdot H)\le P_2(G)\gf(H)$ and $\gm(G)\gf(H)\le\gm(G\cdot H)\le\gm(G)\gm(H)$. These relations are also independent. Relations involving $P_2(G\oplus H)$, $\gf(G\oplus H)$, and $\gm(G\oplus H)$ are examined. An unresolved issue involves a conjecture of Vizing: For all $G$ and $H$, is $\gm(G\oplus H)\ge\gm(G)\gm(H)$?