On Zero-Sum Rado Numbers for the Equation ax_1 + x_2 = x_3

For every positive integer $a$, let $n = R_{ZS}(a)$ be the least integer, provided it exists, such that for every coloring $$\Delta : \{1, 2, ..., n\} \rightarrow \{0, 1, 2\},$$ there exist three integers $x_1, x_2, x_3$ (not necessarily distinct) such that $$\Delta(x_1) + \Delta(x_2) + \Delta(x_3) \equiv 0\ (mod\ 3)$$ and $$ax_1 +x_2 = x_3.$$ If such an integer does not exist, then $R_{ZS}(a) = \infty.$ The main results of this paper are $$R_{ZS}(2) = 12$$ and a lower bound is found for $R_{ZS}(a)$ where $a \geq 2$

On the Factorization of Graphs with Exactly One Vertex of Infinite Degree

AbstractWe give a necessary and sufficient condition for the existence of a 1-factor in graphs with exactly one vertex of infinite degree

Graph Properties in Node-Query Setting: Effect of Breaking Symmetry

The query complexity of graph properties is well-studied when queries are on the edges. We investigate the same when queries are on the nodes. In this setting a graph G = (V,E) on n vertices and a property P are given. A black-box access to an unknown subset S of V is provided via queries of the form "Does i belong to S?". We are interested in the minimum number of queries needed in the worst case in order to determine whether G[S] - the subgraph of G induced on S - satisfies P. Our primary motivation to study this model comes from the fact that it allows us to initiate a systematic study of breaking symmetry in the context of query complexity of graph properties. In particular, we focus on the hereditary graph properties - properties that are closed under deletion of vertices as well as edges. The famous Evasiveness Conjecture asserts that even with a minimal symmetry assumption on G, namely that of vertex-transitivity, the query complexity for any hereditary graph property in our setting is the worst possible, i.e., n. We show that in the absence of any symmetry on G it can fall as low as O(n^{1/(d + 1)}) where d denotes the minimum possible degree of a minimal forbidden sub-graph for P. In particular, every hereditary property benefits at least quadratically. The main question left open is: Can it go exponentially low for some hereditary property? We show that the answer is no for any hereditary property with finitely many forbidden subgraphs by exhibiting a bound of Omega(n^{1/k}) for a constant k depending only on the property. For general ones we rule out the possibility of the query complexity falling down to constant by showing Omega(log(n)*log(log(n))) bound. Interestingly, our lower bound proofs rely on the famous Sunflower Lemma due to Erdos and Rado