Homomorphisms of binary Cayley graphs

A binary Cayley graph is a Cayley graph based on a binary group. In 1982, Payan proved that any non-bipartite binary Cayley graph must contain a generalized Mycielski graph of an odd-cycle, implying that such a graph cannot have chromatic number 3. We strengthen this result first by proving that any non-bipartite binary Cayley graph must contain a projective cube as a subgraph. We further conjecture that any homo- morphism of a non-bipartite binary Cayley graph to a projective cube must be surjective and we prove some special case of this conjecture

On powers of interval graphs and their orders

It was proved by Raychaudhuri in 1987 that if a graph power $G^{k-1}$ is an interval graph, then so is the next power $G^k$. This result was extended to $m$-trapezoid graphs by Flotow in 1995. We extend the statement for interval graphs by showing that any interval representation of $G^{k-1}$ can be extended to an interval representation of $G^k$ that induces the same left endpoint and right endpoint orders. The same holds for unit interval graphs. We also show that a similar fact does not hold for trapezoid graphs.Comment: 4 pages, 1 figure. It has come to our attention that Theorem 1, the main result of this note, follows from earlier results of [G. Agnarsson, P. Damaschke and M. M. Halldorsson. Powers of geometric intersection graphs and dispersion algorithms. Discrete Applied Mathematics 132(1-3):3-16, 2003]. This version is updated accordingl

Sensitivity Lower Bounds from Linear Dependencies

Recently, using the eigenvalue techniques, H. Huang proved that every subgraph of the hypercube of dimension n induced on more than half the vertices has maximum degree at least √ n. Combined with some earlier work, this completed a proof of the sensitivity conjecture. In this work we show how to derive a proof of Huang's result using only linear dependency and independence of vectors associated with the vertices of the hypercube. Our approach leads to several improvements of the result. In particular we prove that in any induced subgraph of H n with more than half the number of vertices, there are two vertices, one of odd parity and the other of even parity, each with at least n vertices at distance at most 2. As an application we show that for any Boolean function f , the polynomial degree of f is bounded above by s 0 (f)s 1 (f), a strictly stronger statement which implies the sensitivity conjecture