A projective-planar signed graph has no two vertex-disjoint negative circles. We prove that every signed graph with no two vertex-disjoint negative circles and no balancing vertex is obtained by taking a projective-planar signed graph or a copy of −K5 and then taking 1, 2, and 3-sums with balanced signed graphs.

Classical spectral clustering is based on a spectral decomposition of a graph Laplacian, obtained from a graph adjacency matrix representing positive graph edge weights describing similarities of graph vertices. In signed graphs, the graph edge weights can be negative to describe disparities of graph vertices, for example, negative correlations in the data. Negative weights lead to possible neg...

Let G be a graph and let c : V (G) → ({1,...,5} 2 ) be an assignment of 2-element subsets of the set {1, . . . , 5} to the vertices of G such that for every edge vw, the sets c(v) and c(w) are disjoint. We call such an assignment a (5, 2)-coloring. A graph is (5,2)-colorable if and only if it has a homomorphism to the Petersen graph. The odd-girth of a graph G is the length of the shortest odd ...

Classical spectral clustering is based on a spectral decomposition of a graph Laplacian, obtained from a graph adjacency matrix representing positive graph edge weights describing similarities of graph vertices. In signed graphs, the graph edge weights can be negative to describe disparities of graph vertices, for example, negative correlations in the data. Negative weights lead to possible neg...

In 1982, Zaslavsky introduced the concept of a proper vertex colouring of a signed graph G as a mapping φ : V (G) → Z such that for any two adjacent vertices u and v the colour φ(u) is different from the colour σ(uv)φ(v), where is σ(uv) is the sign of the edge uv. The substantial part of Zaslavsky’s research concentrated on polynomial invariants related to signed graph colourings rather than on...

Let G be a graph and let c : V (G) → ({1,...,5} 2 ) be an assignment of 2-element subsets of the set {1, . . . , 5} to the vertices of G such that for every edge vw, the sets c(v) and c(w) are disjoint. We call such an assignment a (5, 2)-coloring. A graph is (5,2)-colorable if and only if it has a homomorphism to the Petersen graph. The odd-girth of a graph G is the length of the shortest odd ...

A (1, 2)-eulerian weight w of a cubic graph is called a Hamilton weight if every faithful circuit cover of the graph with respect to w is a set of two Hamilton circuits. Let G be a 3-connected cubic graph containing no Petersen minor. It is proved in this paper that G admits a Hamilton weight if and only if G can be obtained from K4 by a series of4$Y-operations. As a byproduct of the proof of t...