Upper bounds for the 2-hued chromatic number of graphs in terms of the independence number

A 2-hued coloring of a graph $G$ (also known as conditional $(k, 2)$-coloring and dynamic coloring) is a coloring such that for every vertex $v\in V(G)$ of degree at least $2$, the neighbors of $v$ receive at least $2$ colors. The smallest integer $k$ such that $G$ has a 2-hued coloring with $k$ colors, is called the {\it 2-hued chromatic number} of $G$ and denoted by $\chi_2(G)$. In this paper, we will show that if $G$ is a regular graph, then $\chi_{2}(G)- \chi(G) \leq 2 \log _{2}(\alpha(G)) +\mathcal{O}(1)$ and if $G$ is a graph and $\delta(G)\geq 2$, then $\chi_{2}(G)- \chi(G) \leq 1+\lceil \sqrt[\delta -1]{4\Delta^{2}} \rceil ( 1+ \log _{\frac{2\Delta(G)}{2\Delta(G)-\delta(G)}} (\alpha(G)) )$ and in general case if $G$ is a graph, then $\chi_{2}(G)- \chi(G) \leq 2+ \min \lbrace \alpha^{\prime}(G),\frac{\alpha(G)+\omega(G)}{2}\rbrace$.Comment: Dynamic chromatic number; conditional (k, 2)-coloring; 2-hued chromatic number; 2-hued coloring; Independence number; Probabilistic metho

From Cages to Trapping Sets and Codewords: A Technique to Derive Tight Upper Bounds on the Minimum Size of Trapping Sets and Minimum Distance of LDPC Codes

Cages, defined as regular graphs with minimum number of nodes for a given girth, are well-studied in graph theory. Trapping sets are graphical structures responsible for error floor of low-density parity-check (LDPC) codes, and are well investigated in coding theory. In this paper, we make connections between cages and trapping sets. In particular, starting from a cage (or a modified cage), we construct a trapping set in multiple steps. Based on the connection between cages and trapping sets, we then use the available results in graph theory on cages and derive tight upper bounds on the size of the smallest trapping sets for variable-regular LDPC codes with a given variable degree and girth. The derived upper bounds in many cases meet the best known lower bounds and thus provide the actual size of the smallest trapping sets. Considering that non-zero codewords are a special case of trapping sets, we also derive tight upper bounds on the minimum weight of such codewords, i.e., the minimum distance, of variable-regular LDPC codes as a function of variable degree and girth