The odd-valued chromatic polynomial of a signed graph

For a signed graph $\mathrm{\Sigma }=(G,\sigma )$, Zaslavsky defined a proper coloring on Σ and showed that the function counting the number of such colorings is a quasi-polynomial with period two, that is, a pair of polynomials, one for odd values and the other for even values. In this paper, we focus on the case of odd, written as $\chi (\mathrm{\Sigma },x)$ for short. We initially give a homomorphism expression of such colorings, by which the symmetry is considered in counting the number of homomorphisms. Besides, the explicit formulas $\chi (\mathrm{\Sigma },x)$ for some basic classes of signed graphs are presented. As a main result, we give a combinatorial interpretation of the coefficients in $\chi (\mathrm{\Sigma },x)$ and present several applications. In particular, the constant term in $\chi (\mathrm{\Sigma },x)$ gives a new criterion for balancing and a characterization for unbalanced unicyclic graph. At last, we also give a tight bound for the constant term of $\chi (\mathrm{\Sigma },x)$.     

Partial duality and Bollobás and Riordan’s ribbon graph polynomial

Recently S. Chmutov introduced a generalization of the dual of a ribbon graph (or equivalently an embedded graph) and proved a relation between Bollobás and Riordan’s ribbon graph polynomial of a ribbon graph and of its generalized duals. Here I show that the duality relation satisfied by the ribbon graph polynomial can be understood in terms of knot theory and I give a simple proof of the relation which used the homfly polynomial of a knot.     

The signed domatic number of some regular graphs

Let G be a finite and simple graph with vertex set V(G), and let f:V(G)→{−1,1} be a two-valued function. If ∑_x∈N[v]f(x)≥1 for each v∈V(G), where N[v] is the closed neighborhood of v, then f is a signed dominating function on G. A set {f₁,f₂,…,f_d} of signed dominating functions on G with the property that for each x∈V(G), is called a signed dominating family (of functions) on G. The maximum number of functions in a signed dominating family on G is the signed domatic number on G. In this paper, we investigate the signed domatic number of some circulant graphs and of the torus C_p×C_q.     

Circular flow on signed graphs

The circular flow number Φc(G,σ) of a signed graph (G,σ) is the minimum r for which an orientation of (G,σ) admits a circular r-flow. We prove that the circular flow number of a signed graph (G,σ) is equal to the minimum imbalance ratio of an orientation of (G,σ). We then use this result to prove that if G is 4-edge-connected and (G,σ) has a nowhere zero flow, then Φc(G,σ) (as well as Φ(G,σ)) is at most 4. If G is 6-edge-connected and (G,σ) has a nowhere zero flow, then Φc(G,σ) is strictly less than 4.     

On signed cycle domination in graphs

Let G=(V,E) be a graph, a function f:E→{−1,1} is said to be an signed cycle dominating function (SCDF) of G if ∑_e∈E(C)f(e)≥1 holds for any induced cycle C of G. The signed cycle domination number of G is defined as is an SCDF of G}. In this paper, we obtain bounds on , characterize all connected graphs G with , and determine the exact value of for some special classes of graphs G. In addition, we pose some open problems and conjectures.     

Lower bounds on the signed domination numbers of directed graphs

A numerical invariant of directed graphs concerning domination which is named signed domination number γ_S is studied in this paper. We present some sharp lower bounds for γ_S in terms of the order, the maximum degree and the chromatic number of a directed graph.     

The bivariate Ising polynomial of a graph

In this paper we discuss the two variable Ising polynomials in a graph theoretical setting. This polynomial has its origin in physics as the partition function of the Ising model with an external field. We prove some basic properties of the Ising polynomial and demonstrate that it encodes a large amount of combinatorial information about a graph. We also give examples which prove that certain properties, such as the chromatic number, are not determined by the Ising polynomial. Finally we prove that there exist large families of non-isomorphic planar triangulations with identical Ising polynomial.     

On signed graphs with just two distinct adjacency eigenvalues

In this paper, all simple connected signed graphs with maximum degree at most 4 and with just two distinct adjacency eigenvalues are completely characterized, there exists an infinite family of 4-regular signed graphs with just two distinct adjacency eigenvalues.     

On the characterization of trees with signed edge domination numbers 1, 2, 3, or 4

Let G=(V,E) be a simple graph. For an edge e of G, the closed edge-neighbourhood of e is the set N[e]={e^′∈E|e^′ is adjacent to e}∪{e}. A function f:E→{1,−1} is called a signed edge domination function (SEDF) of G if ∑_e^′∈N[e]f(e^′)≥1 for every edge e of G. The signed edge domination number of G is defined as . In this paper, we characterize all trees T with signed edge domination numbers 1, 2, 3, or 4.     

The commutation graph for the longest signed permutation

Using the standard Coxeter presentation for the signed symmetric group ${\mathfrak{S}}_{n+1}^B$ on $n+1$ letters, two reduced expressions for a given signed permutation are in the same commutation class if one expression can be obtained from the other one by applying a finite sequence of commutations. The commutative classes of a given signed permutation can be seen as the vertices of a graph, called the commutation graph, where two classes are connected by an edge if there are elements in those classes that differ by a long braid relation. We define a rank function for the commutation graph for the longest signed permutation, and use this function to compute the diameter and the radius of the graph. We also prove that the commutation graph for the longest signed permutation is not planar for $n>2$, and identify the classes with a single element.     

A simple algorithm to detect balance in signed graphs

We develop a natural correspondence between marked graphs and balanced signed graphs, and exploit it to obtain a simple linear time algorithm by which any signed graph may be tested for balance.     

Every signed planar graph without cycles of length from 4 to 8 is 3-colorable

In this paper, we investigate the signed graph version of Erdös problem: Is there a constant $c$ such that every signed planar graph without $k$-cycles, where $4\le k\le c$, is 3-colorable and prove that each signed planar graph without cycles of length from 4 to 8 is 3-colorable.     

The smallest surface that contains all signed graphs on K4,n

Finding the smallest number of crosscaps that suffice to orientation-embed every edge signature of the complete bipartite graph $K_{m,n}$ is an open problem. In this paper that number for the complete bipartite graph $K_{4,n}$, $n\ge 4$, is determined by using diamond products of signed graphs. The number is $2?\frac{n?1}{2}?+1$, which is attained by $K_{4,n}$ with exactly 1 negative edge, except that when $n=4$, the number is 4, which is attained by $K_{4,4}$ with exactly 4 independent negative edges.