We give here new upper bounds on the size of a smallest feedback vertex set in planar graphs with high girth. In particular, we prove that a planar graph with girth g and size m has a feedback vertex set of size at most 4m 3g , improving the trivial bound of 2m g . We also prove that every 2-connected graph with maximum degree 3 and order n has a feedback vertex set of size at most n+2 3 .

This article presents a method for constructing large girth column-weight 2 QC-LDPC codes. A distance graph is first constructed using an existing method. The distance graph is then converted into a Tanner graph. The proposed method could easily construct codes with girths large than 12 and is more flexible compared to previous methods. Obtained codes show good bit error rate performance compar...

We prove that for all ` ≥ 3 and β > 0 there exists a sparse oriented graph of arbitrarily large order with oriented girth ` and such that any 1/2+β proportion of its arcs induces an oriented cycle of length `. As a corollary we get that there exist infinitely many oriented graphs with vanishing density of oriented girth ` such that deleting any 1/`-fraction of their edges does not destroy all t...

The girth of a graph G is the length of a shortest cycle in G. Dobson (1994, Ph.D. dissertation, Louisiana State University, Baton Rouge, LA) conjectured that every graph G with girth at least 2t+1 and minimum degree at least k t contains every tree T with k edges whose maximum degree does not exceed the minimum degree of G. The conjecture has been proved for t 3. In this paper, we prove Dobson...

For a graph G, the Merrifield-Simmons index i(G) is defined as the total number of independent sets of the graph G. Let G(n, l, k) be the class of unicyclic graphs on n vertices with girth and pendent vertices being resp. l and k. In this paper, we characterize the unique unicyclic graph possessing prescribed girth and pendent vertices with the maximal Merrifield-Simmons index among all graphs ...

We investigate the zero-forcing number for triangle-free graphs. We improve upon the trivial bound, δ ≤ Z(G) where δ is the minimum degree, in the triangle-free case. In particular, we show that 2δ− 2 ≤ Z(G) for graphs with girth of at least 5, and this can be further improved when G has a small cut set. Lastly, we make a conjecture that the lower bound for Z(G) increases as a function of the g...

Nilpotent and idempotent adjacency operator methods are applied to the study of random geometric graphs in a discretized, d-dimensional unit cube [0, 1]. Cycles are enumerated, sizes of maximal connected components are computed, and closed formulas are obtained for graph circumference and girth. Expected numbers of k-cycles, expected sizes of maximal components, and expected circumference and g...

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 ...

We study the problem of recognizing graph powers and computing roots of graphs. We provide a polynomial time recognition algorithm for r-th powers of graphs of girth at least 2r + 3, thus improving a bound conjectured by Farzad et al. (STACS 2009). Our algorithm also finds all r-th roots of a given graph that have girth at least 2r+3 and no degree one vertices, which is a step towards a recent ...

We show a simple deterministic linear time algorithm for computing the minimum spanning tree of graphs on n vertices with girth at least b(t, lg n), where t ≥ 1 is a constant and b(x, y) is a variant of the inverse of the fast-growing Ackermann function. We also prove: (i) A deterministic linear time algorithm for general graphs follows from an algorithm for graphs with girth at least g for any...