Let G be a connected graph of order n. The eccentricity e(v) vertex v is the distance from to farthest v. average mean all eccentricities in G. We give upper bounds on terms n, minimum degree δ, and girth g. In addition, we construct graphs show that, if for given g there exists Moore δ g, then are asymptotically sharp. Moreover, that can improved large Δ.

By a symmetric graph we mean a graph X which automorphism group acts transitively on the arcs of X. A graph is s-regular if its automorphism group acts regularly on the set of its s-arcs. Tutte [31, 32] showed that every finite symmetric cubic graph is s-regular for some s ≤ 5. It is well-known that there are precisely five symmetric cubic graphs of girth less than 6. All these graphs can be re...

A near-polygonal graph is a graph Γ which has a set C of m-cycles for some positive integer m such that each 2-path of Γ is contained in exactly one cycle in C. If m is the girth of Γ , then the graph is called polygonal. We provide a construction of an infinite family of polygonal graphs of arbitrary even girth with 2-arc transitive automorphism groups, showing that there are infinitely many 2...

Let G be a graph and let c : V (G) → ({1,...,5} 2 ) be an assignment of 2-element subsets of the set {1, . . . , 5} to the vertices of G such that for every edge vw, the sets c(v) and c(w) are disjoint. We call such an assignment a (5, 2)-coloring. A graph is (5,2)-colorable if and only if it has a homomorphism to the Petersen graph. The odd-girth of a graph G is the length of the shortest odd ...

Let H be a xed graph. We show that any H-minor free graph of high enough girth has a circular-chromatic number arbitrarily close to two. Equivalently, such graphs have homomorphisms into a large odd circuit. In particular, graphs of high girth and of bounded genus or bounded tree width are \nearly bipartite" in this sense. For example, any planar graph of girth at least 16 admits a homomorphism...

The oriented chromatic number o(H) of an oriented graph H is defined to be the minimum order of an oriented graph H ' such that H has a homomorphism to H' . If each graph in a class ~ has a homomorphism to the same H' , then H ' is ~-universal. Let ~k denote the class of orientations of planar graphs with girth at least k. Clearly, ~3 ~ ~4 ~ ~5... We discuss the existence of ~k-universal graphs...

Let G be a graph and let c : V (G) → ({1,...,5} 2 ) be an assignment of 2-element subsets of the set {1, . . . , 5} to the vertices of G such that for every edge vw, the sets c(v) and c(w) are disjoint. We call such an assignment a (5, 2)-coloring. A graph is (5,2)-colorable if and only if it has a homomorphism to the Petersen graph. The odd-girth of a graph G is the length of the shortest odd ...