42 JANUARY 2004 1053-5888/04/$20.00©2004IEEE e consider the problem of designing unoriented bipartite graphs with large girth. These graphs are the Tanner graphs associated with the parity-check matrix H of low density parity-check (LDPC) codes or Gallager codes. Larger girth improves the computational and bit error rate (BER) performance of these codes. The article overviews several existing m...

In this paper, a recursive algorithm is presented to generate some exponent matrices which correspond to Tanner graphs with girth at least 6. For a J × L exponent matrix E, the lower bound Q(E) is obtained explicitly such that (J, L) QC LDPC codes with girth at least 6 exist for any circulant permutation matrix (CPM) size m ≥ Q(E). The results show that the exponent matrices constructed with ou...

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...

We present a necessary and sufficient condition for a graph of odd-girth 2k + 1 to bound the class of K4-minor-free graphs of odd-girth (at least) 2k + 1, that is, to admit a homomorphism from any such K4-minor-free graph. This yields a polynomial-time algorithm to recognize such bounds. Using this condition, we first prove that every K4-minor free graph of odd-girth 2k+1 admits a homomorphism ...

A series of recent papers shows that it is NP-complete to decide whether an oriented graph admits a homomorphism to the tournament T4 on 4 vertices containing a 4-circuit, each time on a smaller graph class. We improve these results by showing that homomorphism to T4 is NP-complete for bipartite planar subcubic graphs of arbitrarily large fixed girth. We also show that push homomorphism is NP-c...

We give lower bounds on the maximum possible girth of an r-uniform, d-regular hypergraph with at most n vertices, using the definition of a hypergraph cycle due to Berge. These differ from the trivial upper bound by an absolute constant factor (viz., by a factor of between 3/2 + o(1) and 2 + o(1)). We also define a random r-uniform ‘Cayley’ hypergraph on the symmetric group Sn which has girth Ω...

A (δ, g)-cage is a δ-regular graph with girth g and with the least possible number of vertices. In this paper, we show that all (δ, g)-cages with odd girth g ≥ 9 are r-connected, where (r − 1) ≤ δ + √ δ − 2 < r and all (δ, g)-cages with even girth g ≥ 10 are r-connected, where r is the largest integer satisfying r(r−1) 2 4 + 1 + 2r(r − 1) ≤ δ. Those results support a conjecture of Fu, Huang and...

Denote by $\mathcal{G}(n,c,g,k)$ the set of all connected graphs order $n$, having $c$ cycles, girth $g$, and $k$ pendant vertices. In this paper, we give a partial characterisation structure those in maximising number induced subgraphs. For special case where $c=1$, find complete maximal unicyclic graphs. We also derive precise formula for corresponding maximum given following parameters: (1) ...

The focus of this paper is on discussion of a catalog of a class of (3, g) graphs for even girth g. A (k, g) graph is a graph with regular degree k and girth g. This catalog is compared with other known lists of (3, g) graphs such as the enumerations of trivalent symmetric graphs and enumerations of trivalent vertex-transitive graphs, to conclude that this catalog has graphs for more orders tha...