This document provides a brief summary estimate of Galactic free-free emission and Hα emission and their relationship. Particular emphasis is placed on estimating the potential freefree emission in the region of significant confusion for CMB anisotropy measurements. Existing x-ray, ultraviolet and Hα emission provide limits on the radio free-free emission and vice versa. These limits are genera...

We prove that every triangle-free planar graph with minimum degree 3 has radius at least 3; equivalently, no vertex neighborhood is a dominating set.

Every n-vertex graph has two vertices with the same degree (if n ≥ 2). In general, let rep(G) be the maximum multiplicity of a vertex degree in G. An easy counting argument yields rep(G) ≥ n/(2d − 2s + 1), where d is the average degree and s is the minimum degree of G. Equality can hold when 2d is an integer, and the bound is approximately sharp in general, even when G is restricted to be a tre...

The directed power graph G?(G) of a group G is the simple digraph with vertex set such that x?y if y x. G, denoted by G(G), underlying graph. In this paper, for groups and H, following proved. If has no quasicyclic subgroup Cp? which trivial intersection every cyclic K K??Cp?, then G(G)?G(H) implies G?(G)?G?(H). particular, any two torsion-free having isomorphic graphs have graphs.

A vertex v of a graph G is locally connected if the set of neighbors N(v) of v induces a connected subgraph in G. Let B(G) denote the set of vertices of G that are not locally connected. Then G is almost locally connected if B(G) is an independent set and for any x ∈ B(G), there is a vertex y in V (G) \ {x} such that N(x) ∪ {y} induces a connected subgraph of G. The main result of this paper is...

We prove that every planar triangle-free graph on n vertices has fractional chromatic number at most 3− 1 n+1/3 .

We study the conflict-free chromatic number χCF of graphs from extremal and probabilistic point of view. We resolve a question of Pach and Tardos about the maximum conflict-free chromatic number an n-vertex graph can have. Our construction is randomized. In relation to this we study the evolution of the conflict-free chromatic number of the ErdősRényi random graph G(n, p) and give the asymptoti...

Tutte showed that 4-connected planar graphs are Hamiltonian, but it is well known that 3-connected planar graphs need not be Hamiltonian. We show that K2,5-minor-free 3-connected planar graphs are Hamiltonian. This does not extend to K2,5-minor-free 3-connected graphs in general, as shown by the Petersen graph, and does not extend to K2,6-minor-free 3-connected planar graphs, as we show by an i...

A graph H is t-apex if H − X is planar for some set X ⊂ V (H) of size t. For any integer t ≥ 0 and a fixed t-apex graph H, we give a polynomial-time algorithm to decide whether a (t + 3)-connected Hminor-free graph is colorable from a given assignment of lists of size t+4. The connectivity requirement is the best possible in the sense that for every t ≥ 1, there exists a t-apex graph H such tha...

A proportional contact representation of a planar graph is one where each vertex is represented by a simple polygon with area proportional to a given weight and adjacencies between polygons represent edges between the corresponding pairs of vertices. In this paper we study proportional contact representations that use only rectilinear polygons and contain no unused area or hole. There is an alg...