Partitioning a graph into highly connected subgraphs

Given $k\ge 1$, a $k$-proper partition of a graph $G$ is a partition ${\mathcal P}$ of $V(G)$ such that each part $P$ of ${\mathcal P}$ induces a $k$-connected subgraph of $G$. We prove that if $G$ is a graph of order $n$ such that $\delta(G)\ge \sqrt{n}$, then $G$ has a $2$-proper partition with at most $n/\delta(G)$ parts. The bounds on the number of parts and the minimum degree are both best possible. We then prove that If $G$ is a graph of order $n$ with minimum degree $\delta(G)\ge\sqrt{c(k-1)n}$, where $c=\frac{2123}{180}$, then $G$ has a $k$-proper partition into at most $\frac{cn}{\delta(G)}$ parts. This improves a result of Ferrara, Magnant and Wenger [Conditions for Families of Disjoint $k$-connected Subgraphs in a Graph, Discrete Math. 313 (2013), 760--764] and both the degree condition and the number of parts are best possible up to the constant $c$

Preformed lymphocytotoxic antibodies: The effects of class, titer and specificity on liver vs. heart allografts

The effect on liver and heart allograft survival (ACI rats to Lewis rats) was studied after three methods of recipient presensitization and after different intervals between sensitization and transplantation. With comparable lymphocytotoxic antibody titers, liver allografts always survived longer than heart grafts. The titer, class and specificity of the antibodies varied with the method of sensitization. Four skin grafts produced IgG and IgM lymphocytotoxic antibody titers of 1:2,000 to 4,000. The IgG fraction was shown to have hepatic vascular endothelial specificity by indirect immunofluorescence. These primed recipients hyperacutely rejected both heart and liver allografts, which showed vascular deposition of IgG antibodies. Survival of either kind of graft was inversely proportional to the lymphocytotoxic antibody titer and length of time after placement of the last skin graft. Presensitization with a single heterotopic heart graft produced an even higher mixed IgG and IgM lymphocytotoxic antibody titer of 1:8,000 but with less IgG vascular endothelial specificity. These animals also hyperacutely rejected heart or liver grafts with tissue deposition of IgG but less consistently and with a weaker correlation with lymphocytotoxic antibody titers and time after sensitization. Sensitization with two pretransplant blood transfusions produced the lowest titer (1:500 to 1,000) and the least IgG vascular endothelial specificity. Liver allograft survival was routinely enhanced in these animals, and little effect was seen on heart grafts. Collectively, the experiments showed that the liver is not only resistant to antibody‐mediated rejection relative to the heart but is more easily enhanced. A more precise characterization of preformed antibodies may increase the ability to predict the outcome of liver transplantation in sensitized recipients or guide pretransplant strategies to foster enhancing antibodies. Copyright © 1992 American Association for the Study of Liver Disease