Tutte made the conjecture in 1966 that every 2-connected cubic graph not containing the Petersen graph as a minor is 3-edge-colourable. The conjecture is still open, but we show that it is true in general provided it is true for two special kinds of cubic graphs that are almost planar.

A notion of real number graph labelings captures the dependence of the span of an optimal channel assignment on the separations that are required between frequencies assigned to close transmitters. We determine the spans of such optimal labelings for a subfamily of Kneser graphs formed by the complements of the line graphs of complete graphs. This subfamily contains (among others) the Petersen ...

A graph G is signed if each edge is assigned + or −. A signed graph is balanced if there is a bipartition of its vertex set such that an edge has sign − if and only if its endpoints are in different parts. The Edwards-Erdös bound states that every graph with n vertices and m edges has a balanced subgraph with at least m 2 +n−1 4 edges. In the Signed Max Cut Above Tight Lower Bound (Signed Max C...

The matching preclusion number of an even graph is the minimum number of edges whose deletion results in a graph that has no perfect matchings. For many interconnection networks, the optimal sets are precisely those induced by a single vertex. Recently, the conditional matching preclusion number of an even graph was introduced to look for obstruction sets beyond those induced by a single vertex...

We say a graph G is flexibly planar if and only if for every cycle in G, there exists a planar embedding of G such that the cycle bounds a disk whose interior is disjoint from G. We show the complete minor-minimal set of flexibly planar graphs is the single graph K4 We further investigate the property of flexible flatness, a property that has an analogous definition for 3space. We show, with on...

Abstract Network embedding aims to map nodes in a network low-dimensional vector representations. Graph neural networks (GNNs) have received much attention and achieved state-of-the-art performance learning node representation. Using fundamental sociological theories (status theory balance theory) model signed networks, basing GNN on has become hot topic embedding. However, most GNNs fail use e...

Let G = (V , E) be a simple graph on vertex set V and define a function f : V → {−1,1}. The function f is a signed dominating function if for every vertex x ∈ V , the closed neighborhood of x contains more vertices with function value 1 than with −1. The signed domination number of G, γs(G), is the minimum weight of a signed dominating function on G. We give a sharp lower bound on the signed do...

The automorphic A-chromatic index of a graph Γ is the minimum integer m for which Γ has a proper edge-coloring with m colors which is preserved by a given subgroup A of the full automorphism group of Γ. We compute the automorphic A-chromatic index of each generalized Petersen graph when A is the full automorphism group.