arXiv subjects

K. Ehsani

Publications and source records attributed to K. Ehsani.

2 recordsLinked to original sources

arXiv2014

A Note on Signed k-Submatching in Graphs

Let $G$ be a graph of order $n$. For every $v\in V(G)$, let $E_G(v)$ denote the set of all edges incident with $v$. A signed $k$-submatching of $G$ is a function $f:E(G)\longrightarrow \{-1,1\}$, satisfying $f(E_G(v))\leq 1$ for at least $k$ vertices, where $f(S)=\sum_{e\in S}f(e)$, for each $ S\subseteq E(G)$. The maximum of the value of $f(E(G))$, taken over all signed $k$-submatching $f$ of $G$, is called the signed $k$-submatching number and is denoted by $β^k_S(G)$. In this paper, we prove that for every graph $G$ of order $n$ and for any positive integer $k \leq n$, $β^k_S (G) \geq n-k - ω(G)$, where $w(G)$ is the number of components of $G$. This settles a conjecture proposed by Wang. Also, we present a formula for the computation of $β_S^n(G)$.

cs.DM↗

arXiv2014

On the Decision Number of Graphs

Let $G$ be a graph. A good function is a function $f:V(G)\rightarrow \{-1,1\}$, satisfying $f(N(v))\geq 1$, for each $v\in V(G)$, where $ N(v)=\{u\in V(G)\, |\, uv\in E(G) \} $ and $f(S) = \sum_{u\in S} f(u)$ for every $S \subseteq V(G) $. For every cubic graph $G$ of order $ n, $ we prove that $ γ(G) \leq \frac{5n}{7} $ and show that this inequality is sharp. A function $f:V(G)\rightarrow \{-1,1\}$ is called a nice function, if $f(N[v])\le1$, for each $v\in V(G)$, where $ N[v]=\{v\} \cup N(v) $. Define $\overlineβ(G)=max\{f(V(G))\}$, where $f$ is a nice function for $G$. We show that $\overlineβ(G)\ge -\frac{3n}{7}$ for every cubic graph $G$ of order $n$, which improves the best known bound $-\frac{n}{2}$.

cs.DM↗