We exhibit non-switching-isomorphic signed graphs that share a common underlying graph and common chromatic polynomials, thereby answering a question posed by Zaslavsky. We introduce a new pair of bivariate chromatic polynomials that generalises the chromatic polynomials of signed graphs. We establish recursive dominating-vertex deletion formulae for these bivariate chromatic polynomials. As an application, we demonstrate that for a certain family of signed threshold graphs, isomorphism can be characterised by the equality of bivariate chromatic polynomials.
We study extensions of Turán Theorem in edge-weighted settings. A particular case of interest is when constraints on the weight of an edge come from the order of the largest clique containing it. These problems are motivated by Ramsey-Turán type problems. Some of our proofs are based on the method of graph Lagrangians, while the other proofs use flag algebras. Using these results, we prove several new upper bounds on the Ramsey-Turán density of cliques. Other applications of our results are in a recent paper of Balogh, Chen, McCourt, and Murley.
Tutte proved that 4-connected planar graphs are Hamiltonian. It is unknown if there is an analogous result on 1-planar graphs. In this paper, we characterize 4-connected 1-planar chordal graphs and show that all such graphs are Hamiltonian-connected. A crucial tool used in our proof is a characteristic of 1-planar 4-trees.
In this article, we extend the duality relation between face colorings and integer flows of graphs on orientable surfaces in Tutte's flow theory to both orientable and nonorientable surfaces and study Bouchet's 6-flow conjecture from point of embeddings of graphs on surfaces. Consequently, we verify Bouchet's conjecture for a family of embedded graphs, which have a crosscap-contractible circuit.
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