Related papers: Frustration-critical signed graphs

The defective chromatic number of a graph class $\mathcal{G}$ is the minimum integer $k$ such that for some integer $d$, every graph in $\mathcal{G}$ is $k$-colourable such that each monochromatic component has maximum degree at most $d$.…

A signed graph is a pair $(G,\Sigma)$, where $G=(V,E)$ is a graph (in which parallel edges and loops are permitted) with $V=\{1,\ldots,n\}$ and $\Sigma\subseteq E$. The edges in $\Sigma$ are called odd edges and the other edges of $E$ even.…

Given two graphs G and H its 1-{\it join} is the graph obtained by taking the disjoint union of G and H and adding all the edges between a nonempty subset of vertices of G and a nonempty subset of vertices of H. In general, composition…

A connected $k$-chromatic graph $G$ with $k \geq 3$ is said to be triangle-critical, if every edge of $G$ is contained in an induced triangle of $G$ and the removal of any triangle from $G$ decreases the chromatic number of $G$ by three. B.…

A finite simple connected graph $G$ with maximum degree $k$ is $k$-critical if it has chromatic index $\chi'(G)=k+1$ and $\chi'(G-e)=k$ for every edge $e\in E(G)$. Bej and the first author raised the question whether every $k$-critical…

In a graph $G=(V,E)$, a module is a vertex subset $M$ of $V$ such that every vertex outside $M$ is adjacent to all or none of $M$. For example, $\emptyset$, $\{x\}$ $(x\in V )$ and $V$ are modules of $G$, called trivial modules. A graph,…

A graph $G$ is called $k$-factor-critical if $G-S$ has a perfect matching for every $S\subseteq G$ with $|S|=k$. A connected graph $G$ is called $t$-connected if it has more than $t$ vertices and remains connected whenever fewer than $t$…

It is known that, for any $k$-list assignment $L$ of a graph $G$, the number of $L$-list colorings of $G$ is at least the number of the proper $k$-colorings of $G$ when $k>(m-1)/\ln(1+\sqrt{2})$. In this paper, we extend the Whitney's…

A graph $G$ is {$k$-crossing-critical} if $cr(G)\ge k$, but $cr(G\setminus e)<k$ for each edge $e\in E(G)$, where $cr(G)$ is the crossing number of $G$. It is known that for any $k$-crossing-critical graph $G$, $cr(G)\le 2.5k+16$ holds, and…

The presented paper studies the flow number $F(G,\sigma)$ of flow-admissible signed graphs $(G,\sigma)$ with two negative edges. We restrict our study to cubic graphs, because for each non-cubic signed graph $(G,\sigma)$ there is a set…

A graph $G$ of order $n$ is said to be $k$-factor-critical for integers $1\leq k < n$, if the removal of any $k$ vertices results in a graph with a perfect matching. A $k$-factor-critical graph $G$ is called minimal if for any edge $e\in…

A graph $G$ is $k$-{\em critical} if it has chromatic number $k$, but every proper subgraph of $G$ is $(k-1)$--colorable. Let $f_k(n)$ denote the minimum number of edges in an $n$-vertex $k$-critical graph. Recently the authors gave a lower…

A complex unit gain graph ($\mathbb{T}$-gain graph), $\Phi = (G, \varphi)$ is a graph where the function $\varphi$ assigns a unit complex number to each orientation of an edge of $G$, and its inverse is assigned to the opposite orientation.…

A $k$-matching in a graph $G$ is defined as a function $f:E(G) \rightarrow \{0,1,\ldots,k\}$ satisfying $\sum_{e\in E_G(v)} f(e)$ $\leq k$ for each vertex $v\in V(G)$, where $E_G(v)$ denotes the set of edges incident to $v$ in $G$. For…

We study chip-firing on a signed graph $G_\phi$, employing a general theory of chip-firing on invertible matrices introduced by Guzm\'an and Klivans. Here a negative edge designates an adversarial relationship, so that firing a vertex…

A $k$-$\gamma_{c}$-edge critical graph is a graph $G$ with the connected domination number $\gamma_{c}(G) = k$ and $\gamma_{c}(G + uv) < k$ for every $uv \in E(\overline{G})$. Further, a $2$-connected graph $G$ is said to be…

Assume $G$ is a graph. We view $G$ as a symmetric digraph, in which each edge $uv$ of $G$ is replaced by a pair of opposite arcs $e=(u,v)$ and $e^{-1}=(v,u)$. Assume $S$ is an inverse closed subset of permutations of positive integers. We…

A graph $G$ is $k$-factor-critical if $G-S$ has a perfect matching for any $k$-subset $S$ of the vertex set of $G$. In this paper, we investigate the factor-criticality of graphs with fixed minimum degree and provide sufficient conditions…

A graph $G$ is \textit{$k$-critical} if $\chi(G) = k$ and every proper subgraph of $G$ is $(k - 1)$-colorable, and if $L$ is a list-assignment for $G$, then $G$ is \textit{$L$-critical} if $G$ is not $L$-colorable but every proper induced…

We analyse signed networks from the perspective of balance theory which predicts structural balance as a global structure for signed social networks that represent groups of friends and enemies. The scarcity of balanced networks encouraged…