The inapproximability for the (0,1)-additive number

An {\it additive labeling} of a graph $G$ is a function $\ell :V(G) \rightarrow\mathbb{N}$, such that for every two adjacent vertices $v$ and $u$ of $G$, $\sum_{w \sim v}\ell(w)\neq \sum_{w \sim u}\ell(w)$ ($x \sim y$ means that $x$ is joined to $y$). The {\it additive number} of $G$, denoted by $\eta(G)$, is the minimum number $k$ such that $G$ has a additive labeling $\ell :V(G) \rightarrow \mathbb{N}_k$. The {\it additive choosability} of a graph $G$, denoted by $\eta_{\ell}(G)$, is the smallest number $k$ such that $G$ has an additive labeling for any assignment of lists of size $k$ to the vertices of $G$, such that the label of each vertex belongs to its own list. Seamone (2012) \cite{a80} conjectured that for every graph $G$, $\eta(G)= \eta_{\ell}(G)$. We give a negative answer to this conjecture and we show that for every $k$ there is a graph $G$ such that $\eta_{\ell}(G)- \eta(G) \geq k$. A {\it $(0,1)$-additive labeling} of a graph $G$ is a function $\ell :V(G) \rightarrow\{0,1\}$, such that for every two adjacent vertices $v$ and $u$ of $G$, $\sum_{w \sim v}\ell(w)\neq \sum_{w \sim u}\ell(w)$. A graph may lack any $(0,1)$-additive labeling. We show that it is $\mathbf{NP}$-complete to decide whether a $(0,1)$-additive labeling exists for some families of graphs such as perfect graphs and planar triangle-free graphs. For a graph $G$ with some $(0,1)$-additive labelings, the $(0,1)$-additive number of $G$ is defined as $\sigma_{1} (G) = \min_{\ell \in \Gamma}\sum_{v\in V(G)}\ell(v)$ where $\Gamma$ is the set of $(0,1)$-additive labelings of $G$. We prove that given a planar graph that admits a $(0,1)$-additive labeling, for all $\varepsilon >0$, approximating the $(0,1)$-additive number within $n^{1-\varepsilon}$ is $\mathbf{NP}$-hard.Comment: 14 pages, 3 figures, Discrete Mathematics & Theoretical Computer Scienc

(2/2/3)(2/2/3)-SAT problem and its applications in dominating set problems

The satisfiability problem is known to be $\mathbf{NP}$-complete in general and for many restricted cases. One way to restrict instances of $k$-SAT is to limit the number of times a variable can be occurred. It was shown that for an instance of 4-SAT with the property that every variable appears in exactly 4 clauses (2 times negated and 2 times not negated), determining whether there is an assignment for variables such that every clause contains exactly two true variables and two false variables is $\mathbf{NP}$-complete. In this work, we show that deciding the satisfiability of 3-SAT with the property that every variable appears in exactly four clauses (two times negated and two times not negated), and each clause contains at least two distinct variables is $\mathbf{NP}$-complete. We call this problem $(2/2/3)$-SAT. For an $r$-regular graph $G = (V,E)$ with $r\geq 3$, it was asked in [Discrete Appl. Math., 160(15):2142--2146, 2012] to determine whether for a given independent set $T$ there is an independent dominating set $D$ that dominates $T$ such that $T \cap D =\varnothing$? As an application of $(2/2/3)$-SAT problem we show that for every $r\geq 3$, this problem is $\mathbf{NP}$-complete. Among other results, we study the relationship between 1-perfect codes and the incidence coloring of graphs and as another application of our complexity results, we prove that for a given cubic graph $G$ deciding whether $G$ is 4-incidence colorable is $\mathbf{NP}$-complete

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