Using voltage graphs, we construct the the smallest known trivalent graph of girth 16. The graph has 960 vertices and is a lift of the Petersen graph. By way of preparation, other lifts of the Petersen graph are discussed. AMS Subject Classifications: 05C25, 05C35

Signed link prediction in social networks aims to reveal the underlying relationships (i.e., links) among users nodes) given their existing positive and negative interactions observed. Most of prior efforts are devoted learning node embeddings with graph neural (GNNs), which preserve signed network topology by message-passing along edges facilitate downstream task. Nevertheless, graph-based app...

In my doctoral thesis [1], I claimed a proof that the branchwidth of the Petersen graph had branchwidth 5 and proceeded to prove that it was a member of the obstruction set for graphs with branchwidth at most 4. I have stated this claim in Hicks [2] and Hicks et al. [4]. There was, however, an error in the construction of the tangle used to prove the claim. The branchwidth of the Petersen graph...

Tutte [W.T. Tutte, On the algebraic theory of graph colorings, J. Combin. Theory 1 (1966) 15–20] conjectured that every bridgeless Petersen-minor free graph admits a nowhere-zero 4-flow. Let (P10)μ̄ be the graph obtained from the Petersen graph by contracting μ edges from a perfect matching. In this paper we prove that every bridgeless (P10)3̄-minor free graph admits a nowhere-zero 4-flow. c © 20...

This paper surveys recent development of concepts related to coloring signed graphs. Various approaches are presented and discussed.

A complete classification is given of vertex primitive and vertex bi-primitive s-arc regular graphs with s ≥ 3. In particular, it is shown that the Petersen graph and Coxeter graph are the only vertex primitive 3-arc regular graphs, and that vertex bi-primitive 3-arc regular graphs consist of the complete bipartite graph K3,3, the standard double covers of the Petersen graph and Coxeter graph, ...