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We prove a lower bound of Ω ( d 3 / 2 ⋅ ( 2 / 3 ) d ) on the kissing number in dimension d . This improves the classical lower bound of Chabauty, Shannon, and Wyner by a linear factor in the dimension. We obtain a similar linear factor improvement to the best known lower bound on the maximal size of a spherical code of acute angle θ in high dimensions. [ABSTRACT FROM AUTHOR]

A key feature of Simonovits' proof of the classic Erdőos--Pósa theorem is a simple subgraph of the host graph, a frame, that determines the outcome of the theorem. We transfer this frame technique to A-paths. With it we deduce a simple proof of Gallai's theorem, although with a worse bound, and we verify the Erdős--Pósa property for long and for even A-paths. We also show that even A-paths do not have the edge-Erdős--Pósa property. [ABSTRACT FROM AUTHOR]

The maximum number vertices of a graph G inducing a 2-regular subgraph of G is denoted by $$c_\mathrm{ind}(G)$$ . We prove that if G is an r-regular graph of order n, then $$c_\mathrm{ind}(G) \ge \frac{n}{2(r-1)} + \frac{1}{(r-1)(r-2)}$$ and we prove that if G is a cubic, claw-free graph on order n, then $$c_\mathrm{ind}(G) > \frac{13}{20}n$$ and this bound is asymptotically best possible. [ABSTRACT FROM AUTHOR]

Abstract: The domination number and the total domination number of a graph without an isolated vertex are among the most well-studied parameters in graph theory. While the inequality is an almost immediate consequence of the definition, the extremal graphs for this inequality are not well understood. Furthermore, even very strong additional assumptions do not allow us to improve the inequality by much. In the present paper we consider the relation of and for cubic graphs of large girth. Clearly, in this case is at least where is the order of . If is close to , then this forces a certain structure within . We exploit this structure and prove an upper bound on , which depends on the value of . As a consequence, we can considerably improve the inequality for cubic graphs of large girth. [Copyright &y& Elsevier]