Related papers: Maximum Frustration in Signed Generalized Petersen…

A signed circuit is a minimal signed graph (with respect to inclusion) that admits a nowhere-zero flow. We show that each flow-admissible signed graph on $m$ edges can be covered by signed circuits of total length at most $(3+2/3)\cdot m$,…

A signed graph $(G,\sigma)$ on $n$ vertices is called a \textit{parity signed graph} if there is a bijective mapping $f \colon V(G) \rightarrow \{1,\ldots,n\}$ such that $f(u)$ and $f(v)$ have same parity if $\sigma(uv)=1$, and opposite…

Signed networks with positive and negative interaction are widely observed in the real systems. The negative links would induce frustration, then affect global properties of the system. Based on previous studies, frustration of signed…

A graph $G$ is called $C_k$-saturated if $G$ is $C_k$-free but $G+e$ not for any $e\in E(\overline{G})$. The saturation number of $C_k$, denoted $sat(n,C_k)$, is the minimum number of edges in a $C_k$-saturated graph on $n$ vertices.…

For a graph $G$, let $c_k(G)$ be the number of spanning trees of $G$ with maximum degree at most $k$. For $k \ge 3$, it is proved that every connected $n$-vertex $r$-regular graph $G$ with $r \ge \frac{n}{k+1}$ satisfies $$ c_k(G)^{1/n} \ge…

We study homomorphism problems of signed graphs. A signed graph is an undirected graph where each edge is given a sign, positive or negative. An important concept for signed graphs is the operation of switching at a vertex, which is to…

For a connected graph $G$, let $\mu(G)$ denote the distance spectral radius of $G$. A matching in a graph $G$ is a set of disjoint edges of $G$. The maximum size of a matching in $G$ is called the matching number of $G$, denoted by…

We determine the maximum index and the signed graphs with the maximum index among all $\mathcal{K}_{2,t}^-$-free unbalanced signed graphs with fixed order for $t\geq 3$, as well as the second maximum index and the signed graphs with the…

Let $H$ be a fixed graph. We say that a graph $G$ is $H$-saturated if it has no subgraph isomorphic to $H$, but the addition of any edge to $G$ results in an $H$-subgraph. The saturation number $\mathrm{sat}(H,n)$ is the minimum number of…

Let $\Gamma=(G,\sigma)$ be a signed graph and $A(G,\sigma)$ be its adjacency matrix. Denote by $gr(G)$ the girth of $G$, which is the length of the shortest cycle in $G$. Let $r(G,\sigma)$ be the rank of $(G,\sigma)$. In this paper, we will…

We consider three extremal problems about the number of copies of a fixed graph in another larger graph. First, we correct an error in a result of Reiher and Wagner and prove that the number of $k$-edge stars in a graph with density $x \in…

A graph drawn in the plane is called k-quasi-planar if it does not contain k pairwise crossing edges. It has been conjectured for a long time that for every fixed k, the maximum number of edges of a k-quasi-planar graph with n vertices is…

Given a graph $G=(V,E)$ and a set of vertices marked as filled, we consider a color-change rule known as zero forcing. A set $S$ is a zero forcing set if filling $S$ and applying all possible instances of the color change rule causes all…

A $1$-factorization of a graph $G$ is a collection of edge-disjoint perfect matchings whose union is $E(G)$. A trivial necessary condition for $G$ to admit a $1$-factorization is that $|V(G)|$ is even and $G$ is regular; the converse is…

An $acyclic$ edge coloring of a graph is a proper edge coloring such that there are no bichromatic cycles. The \emph{acyclic chromatic index} of a graph is the minimum number k such that there is an acyclic edge coloring using k colors and…

The paper studies edge-coloring of signed multigraphs and extends classical Theorems of Shannon and K\"onig to signed multigraphs. We prove that the chromatic index of a signed multigraph $(G,\sigma_G)$ is at most $\lfloor \frac{3}{2}…

We examine the conditions under which a signed graph contains an edge or a vertex that is contained in a unique negative circle or a unique positive circle. For an edge in a unique signed circle, the positive and negative case require the…

A graph is $(d_1, ..., d_r)$-colorable if its vertex set can be partitioned into $r$ sets $V_1, ..., V_r$ so that the maximum degree of the graph induced by $V_i$ is at most $d_i$ for each $i\in \{1, ..., r\}$. For a given pair $(g, d_1)$,…

An axis-parallel $d$-dimensional box is a cartesian product $I_1\times I_2\times \dots \times I_b$ where $I_i$ is a closed sub-interval of the real line. For a graph $G = (V,E)$, the $boxicity \ of \ G$, denoted by $\text{box}(G)$, is the…

A signed graph is a graph together with an assignment of signs to the edges. A closed walk in a signed graph is said to be positive (negative) if it has an even (odd) number of negative edges, counting repetition. Recognizing the signs of…