Jeager et al introduced a concept of group connectivity as an generalization of nowhere zero flows and its dual concept group coloring, and conjectured that every 5-edge connected graph is Z3-connected. For planar graphs, this is equivalent to that every planar graph with girth at least 5 must have group chromatic number at most 3. In this paper we show that if G is a plane graph with girth at ...

A connected graph of girth m ≥ 3 is called a polygonal graph if it contains a set of m-gons such that every path of length two is contained in a unique element of the set. In this paper we investigate polygonal graphs of girth 6 or more having automorphism groups which are transitive on the vertices and such that the vertex stabilizers are 3-homogeneous on adjacent vertices. We previously showe...

The odd-girth of a graph is the length of a shortest odd circuit. A conjecture by Pavol Hell about circular coloring is solved in this article by showing that there is a function f( ) for each : 0 < < 1 such that, if the odd-girth of a planar graph G is at least f( ), then G is (2 + )-colorable. Note that the function f( ) is independent of the graph G and → 0 if and only if f( )→∞. A key lemma...

A star-factor of a graph G is a spanning subgraph of G such that each component of which is a star. Recently, Hartnell and Rall studied a family U of graphs satisfying the property that every star-factor of a member graph has the same number of edges. They determined the family U when the girth is at least five. In this paper, we investigate the family of graphs with girth three and determine a...

We prove that a connected graph of diameter at least 4 and of girth 7 or more (in particular, a tree) can be exactly reconstructed from metric balls of radius 2 of all its vertices. On the other hand, there exist graphs of diameter 3 and of girth 6 which are not reconstructible. This new graph theory problem is motivated by reconstruction of chemical compounds. © 2007 Published by Elsevier B.V.

A proper vertex coloring of a graph G is equitable if the sizes of color classes differ by at most one. The equitable chromatic threshold χeq(G) of G is the smallest integer m such that G is equitably n-colorable for all n ≥ m. We show that for planar graphs G with minimum degree at least two, χeq(G) ≤ 4 if the girth of G is at least 10, and χeq(G) ≤ 3 if the girth of G is at least 14.

Tian and Meng in [Y. Tian and J. Meng, c  -Optimally half vertex transitive graphs with regularity , Information Processing Letters 109 (2009) 683-686] shown that a connected half vertex transitive graph with regularity and girth is cyclically optimal. In this paper, we show that a connected half vertex transitive graph G is super cyclically edge-connected if minimum degree k k   6 g G   ...

This article studies the girth and cogirth problems for connected matroids. The problem of finding the cogirth of a graphic matroid has been intensively studied, but studies on the equivalent problem for a linear matroid or a general matroid have been rarely reported. Based on the duality and connectivity of a matroid, we prove properties associated with the girth and cogirth of a matroid whose...

We have been investigating the cryptographical properties of infinite families of simple graphs of large girth with the special colouring of vertices during the last 10 years. Such families can be used for the development of cryptographical algorithms (on symmetric or public key modes) and turbocodes in error correction theory. Only few families of simple graphs of large unbounded girth and arb...

In this paper, the concept of the broken diagonal pair in the chess-like square board is used to define some well-structured block designs whose incidence matrices can be considered as the parity-check matrices of some high rate cycle codes with girth 12. Interestingly, the constructed regular cycle codes with row-weights t, 3 ≤ t ≤ 20, t 6= 7, 15, 16, have the best lengths among the known regu...