Design of Nonbinary LDPC Cycle Codes with Large Girth from Circulants and Finite Fields

In this paper, we study a class of nonbinary LDPC (NBLDPC) codes whose parity-check matrices have column weight 2, called NBLDPC cycle codes. We propose a design framework of 2 , ρ -regular binary quasi-cyclic (QC) LDPC codes and then construct NBLDPC cycle codes of large girth based on circulants and finite fields by randomly choosing the nonzero field elements in their parity-check matrices. For enlarging the girth values, our approach is twofold. First, we give an exhaustive search of circulants with column/row weight ρ and design a masking matrix with good cycle distribution based on the edge-node relation in undirected graphs. Second, according to the designed masking matrix, we construct the exponent matrix based on finite fields. The iterative decoding performances of the constructed codes on the additive white Gaussian noise (AWGN) channel are finally provided.

Box and Segment Intersection Graphs with Large Girth and Chromatic Number

We prove that there are intersection graphs of axis-aligned boxes in R3 and intersection graphs of straight lines in R3 that have arbitrarily large girth and chromatic number.

Circular Chromatic Number of Signed Graphs

A signed graph is a pair $(G, \sigma)$, where $G$ is a graph (loops and multi edges allowed) and $\sigma: E(G) \to \{+, -\}$ is a signature which assigns to each edge of $G$ a sign. Various notions of coloring of signed graphs have been studied. In this paper, we extend circular coloring of graphs to signed graphs. Given a signed graph $(G, \sigma)$ with no positive loop, a circular $r$-coloring of $(G, \sigma)$ is an assignment $\psi$ of points of a circle of circumference $r$ to the vertices of $G$ such that for every edge $e=uv$ of $G$, if $\sigma(e)=+$, then $\psi(u)$ and $\psi(v)$ have distance at least $1$, and if $\sigma(e)=-$, then $\psi(v)$ and the antipodal of $\psi(u)$ have distance at least $1$. The circular chromatic number $\chi_c(G, \sigma)$ of a signed graph $(G, \sigma)$ is the infimum of those $r$ for which $(G, \sigma)$ admits a circular $r$-coloring. For a graph $G$, we define the signed circular chromatic number of $G$ to be $\max\{\chi_c(G, \sigma): \sigma \text{ is a signature of $G$}\}$. We study basic properties of circular coloring of signed graphs and develop tools for calculating $\chi_c(G, \sigma)$. We explore the relation between the circular chromatic number and the signed circular chromatic number of graphs, and present bounds for the signed circular chromatic number of some families of graphs. In particular, we determine the supremum of the signed circular chromatic number of $k$-chromatic graphs of large girth, of simple bipartite planar graphs, $d$-degenerate graphs, simple outerplanar graphs and series-parallel graphs. We construct a signed planar simple graph whose circular chromatic number is $4+\frac{2}{3}$. This is based and improves on a signed graph built by Kardos and Narboni as a counterexample to a conjecture of Máčajová, Raspaud, and Škoviera.

Ramsey-type numbers involving graphs and hypergraphs with large girth

Erdős asked if, for every pair of positive integers g and k, there exists a graph H having girth (H) = k and the property that every r-colouring of the edges of H yields a monochromatic cycle C k . The existence of such graphs H was confirmed by the third author and Ruciński. We consider the related numerical problem of estimating the order of the smallest graph H with this property for given integers r and k. We show that there exists a graph H on R10k2; k15k3 vertices (where R = R(C k ; r) is the r-colour Ramsey number for the cycle C k ) having girth (H) = k and the Ramsey property that every r-colouring of the edges of H yields a monochromatic C k Two related numerical problems regarding arithmetic progressions in subsets of the integers and cliques in graphs are also considered.

Fractional DP-chromatic number of planar graphs of large girth

This paper proves that for any integer [Formula: see text], every planar graph [Formula: see text] of girth at least [Formula: see text] has fractional DP-chromatic number at most [Formula: see text].

Uniquely $D$-Colourable Digraphs with Large Girth II: Simplification via Generalization

We prove that for every digraph $D$ and every choice of positive integers $k$, $\ell$ there exists a digraph $D^*$ with girth at least $\ell$ together with a surjective acyclic homomorphism $\psi\colon D^*\to D$ such that: (i) for every digraph $C$ of order at most $k$, there exists an acyclic homomorphism $D^*\to C$ if and only if there exists an acyclic homomorphism $D\to C$; and (ii) for every $D$-pointed digraph $C$ of order at most $k$ and every acyclic homomorphism $\varphi\colon D^*\to C$ there exists a unique acyclic homomorphism $f\colon D\to C$ such that $\varphi=f\circ\psi$. This implies the main results in [A. Harutyunyan et al., Uniquely $D$-colourable digraphs with large girth, Canad. J. Math., 64(6) (2012), 1310—1328; MR2994666] analogously with how the work [J. Nešetřil and X. Zhu, On sparse graphs with given colorings and homomorphisms, J. Combin. Theory Ser. B, 90(1) (2004), 161—172; MR2041324] generalizes and extends [X. Zhu, Uniquely $H$-colorable graphs with large girth, J. Graph Theory, 23(1) (1996), 33—41; MR1402136].