Let q be a prime a power and k an integer such that 3 ≤ k ≤ q. In this paper we present a method using Latin squares to construct adjacency matrices of k-regular bipartite graphs of girth 8 on 2(kq − q) vertices. Some of these graphs have the smallest number of vertices among the known regular graphs with girth 8.

We prove that, the acyclic chromatic index a′(G) ≤ 6∆ for all graphs with girth at least 9. We extend the same method to obtain a bound of 4.52∆ with the girth requirement g ≥ 220. We also obtain a relationship between g and a′(G).

A star coloring of a graph is a proper vertex-coloring such that no path on four vertices is 2-colored. We prove that the vertices of every planar graph of girth 6 (respectively 7,8) can be star colored from lists of size 8 (respectively 7,6). We give an example of a planar graph of girth 5 that requires 6 colors to star color.

This works presents a formalization of the Girth-Chromatic number theorem in graph theory, stating that graphs with arbitrarily large girth and chromatic number exist. The proof uses the theory of Random Graphs to prove the existence with probabilistic arguments and is based on [1].

We study the asymptotic value of several extremal problems on graphs and hypergraphs, that arise as generalized notions of girth. Apart from being combinatorially natural questions, they are motivated by computational-theoretic applications. 1. An `-subgraph is a subgraph with ` edges per vertex, or equivalently, average degree 2`. What is the optimal upper bound S`(n, d), such that any graph o...

Reiman’s inequality for the size of bipartite graphs of girth six is generalised to girth eight. It is optimal in as far as it admits the algebraic structure of generalised quadrangles as case of equality. This enables us to obtain the optimal estimate e ∼ v for balanced bipartite graphs. We also get an optimal estimate for very unbalanced graphs.

The girth of graphs on Weyl groups, with no restriction on the associated root system, is determined. It is shown that the girth, when it is deened, is 3 except for at most four graphs for which it does not exceed 4.

In this paper, we deal with zero-divisor graphs of posets. We prove that the diameter of such a graph is either 1, 2 or 3 while its girth is either 3, 4 or ∞. We also characterize zero-divisor graphs of posets in terms of their diameter and girth. © 2012 Elsevier B.V. All rights reserved.

The relation between the girth and the error correction capability of column-weight-three LDPC codes is investigated. Specifically, it is shown that the Gallager A algorithm can correct g/2 − 1 errors in g/2 iterations on a Tanner graph of girth g ≥ 10.

The Girth-Chromatic number theorem is a theorem from graph theory, stating that graphs with arbitrarily large girth and chromatic number exist. We formalize a probabilistic proof of this theorem in the Isabelle/HOL theorem prover, closely following a standard textbook proof and use this to explore the use of the probabilistic method in a theorem prover.