Related papers: Circuit Covers of Signed Eulerian Graphs
A frame matroid M is graphic if there is a graph G with cycle matroid isomorphic to M. In general, if there is one such graph, there will be many. Zaslavsky has shown that frame matroids are precisely those having a representation as a…
Due to Veblen's Theorem, if a connected multigraph $X$ has even degrees at each vertex, then it is Eulerian and its edge set has a partition into cycles. In this paper, we show that an Eulerian multigraph has a unique partition into cycles…
In this paper we describe a physical problem, based on electromagnetic fields, whose topological constraints are higher dimensional versions of Kirchhoff's laws, involving $2-$ simplicial complexes embedded in $\mathbb{R} ^3$ rather than…
A graph $G$ is well-covered if it has no isolated vertices and all the maximal independent sets have the same cardinality. If furthermore two times this cardinality is equal to $|V(G)|$, the graph $G$ is called very well-covered. The class…
By coloring a signed graph by signed colors, one obtains the signed chromatic polynomial of the signed graph. For each signed graph we construct graded cohomology groups whose graded Euler characteristic yields the signed chromatic…
In this paper, we study the Tur\'{a}n problem of signed graphs version. Suppose that $\dot{G}$ is a connected unbalanced signed graph of order $n$ with $e(\dot{G})$ edges and $e^-(\dot{G})$ negative edges, and let $\rho(\dot{G})$ be the…
A {\em rooted graph} is a graph together with a designated vertex subset, called the {\em roots}. In this paper, we consider rooted graphs embedded in a fixed surface. A collection of faces of the embedding is a {\em face cover} if every…
In a closed 2-cell embedding of a graph each face is homeomorphic to an open disk and is bounded by a cycle in the graph. The Orientable Strong Embedding Conjecture says that every 2-connected graph has a closed 2-cell embedding in some…
The Cycle double cover (CDC) conjecture states that for every bridgeless graph $G$, there exists a family $\mathcal{F}$ of cycles such that each edge of the graph is contained in exactly two members of $\mathcal{F}$. Given an embedding of a…
In a signed graph each edge has a sign, $+1$ or $-1$. We introduce in the present paper a new definition of connection in a signed graph by the existence of both positive and negative chains between vertices. We prove some results and…
A graph in which every connected induced subgraph has a disconnected complement is called a cograph. Such graphs are precisely the graphs that do not have the 4-vertex path as an induced subgraph. We define a $2$-cograph to be a graph in…
We introduce a new class of matroids, called graph curve matroids. A graph curve matroid is associated to a graph and defined on the vertices of the graph as a ground set. We prove that these matroids provide a combinatorial description of…
A signed graph $ (G, \Sigma)$ is a graph positive and negative ($\Sigma $ denotes the set of negative edges). To re-sign a vertex $v$ of a signed graph $ (G, \Sigma)$ is to switch the signs of the edges incident to $v$. If one can obtain $…
The edges surrounding a face of a map $M$ form a cycle $C$, called the boundary cycle of the face, and $C$ is often not a simple cycle. If the map $M$ is arc-transitive, then there is a cyclic subgroup of automorphisms of $M$ which leaves…
The Hoffman program with respect to any real or complex square matrix $M$ associated to a graph $G$ stems from Hoffman's pioneering work on the limit points for the spectral radius of adjacency matrices of graphs does not exceed…
Take a graph $G$, an edge subset $\Sigma\subseteq E(G)$, and a set of terminals $T\subseteq V(G)$ where $|T|$ is even. The triple $(G,\Sigma,T)$ is called a signed graft. A $T$-join is odd if it contains an odd number of edges from…
Odd coloring is a proper coloring with an additional restriction that every non-isolated vertex has some color that appears an odd number of times in its neighborhood. The minimum number of colors $k$ that can ensure an odd coloring of a…
We say that a signed graph is $k$-critical if it is not $k$-colorable but every one of its proper subgraphs is $k$-colorable. Using the definition of colorability due to Naserasr, Wang, and Zhu that extends the notion of circular…
A signed graph is a pair $(G,\sigma)$, where $G$ is a graph and $\sigma: E(G)\rightarrow \{-, +\}$, called signature, is an assignment of signs to the edges. Given a signed graph $(G,\sigma)$ with no negative loops, a balanced…
In this paper, we define the Mycielskian of a signed graph and discuss the properties of balance and switching in the Mycielskian of a given signed graph. We provide a condition for ensuring the Mycielskian of a balanced signed graph…