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For an edge $uv$ of a finite simple graph $G$, its imbalance is $|d_G(u)-d_G(v)|$, and the imbalance multiset $M_G$ consists of the imbalances of all edges of $G$. Kozerenko and Skochko conjectured that $M_G$ is graphic whenever every edge has positive imbalance. We prove this conjecture. The main ingredient is the fol...
For a finite simple graph $G$ with at least one edge, the bondage number $b(G)$ is the least number of edges whose deletion increases the domination number $\gamma(G)$. Teschner conjectured that $b(G)\le \tfrac32\Delta(G)$ for every graph $G$. We disprove this conjecture by giving a connected cubic bipartite graph on e...
For an edge $uv$ of a finite simple graph $G$, its imbalance is $|d_G(u)-d_G(v)|$, and the imbalance multiset $M_G$ consists of the imbalances of all edges of $G$. Kozerenko and Skochko conjectured that $M_G$ is graphic whenever every edge has positive imbalance. We prove this conjecture. The main ingredient is the fol...
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