Atsuhiro Nakamoto

Title: Bad circuit graphs in Hamiltonian-like problems

Tutte proved that every \( 4 \)-connected planar graph is Hamiltonian, but there are non-Hamiltonian \( 3 \)-connected planar graphs. We can find many research on Hamiltonian-like properties of \( 3 \)-connected planar graphs. In solving such problems, we often encounter a face subdivision of a triangulation as one with a bad structure. In my talk, dealing with circuit graphs (which are similar to \( 3 \)-connected planar graphs), we show that in some of the above problems, only the face subdivisions of a triangulation have the bad structures. We also discuss an extension of these results to a higher surface. This is a joint work with Kenta Ozeki and Daiki Takahashi.

Karel Devriendt

Title: Graphs with nonnegative resistance curvature

In recent work, we introduced a new class of graphs based on discrete curvature. A connected graph is called "resistance nonnegative" (RN) if there exists a distribution on its spanning trees, such that every vertex has expected degree at most 2 in a random spanning tree under this distribution. Some examples are vertex-transitive and Hamiltonian graphs, some non-examples are even graphs without a perfect matching. In this talk, I will discuss some initial results on these graphs and mention a number of open problems.

Yelena Yuditsky

Title: Polynomial Gyárfás-Sumner conjecture for graphs of bounded boxicity

We prove that for every positive integer \( d \) and forest \( F \), the class of intersection graphs of axis-aligned boxes in \( \mathbb{R}^d \) with no induced \( F \) subgraph is(polynomially) \( \chi \)-bounded. This is a joint work with James Davies.

Kenta Ozeki

Title: Colouring normal quadrangulations of projective spaces

Youngs proved that all nonbipartite quadrangulations of the projective plane do not have a \( 3 \)-coloring. In this talk, we extend this to quadrangulations of projective spaces in higher dimension. In addition, we also discuss colorings of such quadrangulations. This is a joint work with Kaiser, Lo, Nozaki and Nakamoto.

Jozefien D'haeseleer

Title: Cospectral mates for generalized Johnson and Grassmann graphs

A major problem in the field of spectral graph theory is to decide whether a given graph is determined by its spectrum (the eigenvalues of its adjacency matrix). This problem has been solved for many families of graphs; sometimes by proving that the spectrum determines the graph, and sometimes by constructing cospectral mates (non-isomorphic graphs with the same spectrum). Although a conjecture by W. Haemers states that almost all graphs are determined by their spectrum, many well-known graphs seem to have cospectral mates. This is, in particular, the case for several graphs in the Johnson and Grassmann association schemes. In this talk, I will present recent work, in which we contribute to this line of research by constructing cospectral mates for a number of open cases in the Johnson and Grassmann schemes. For this, we use switching techniques due to Godsil and McKay as well as Wang, Qiu and Hu.

Shunichi Maezawa

Title: Reconfiguration graphs induced by rainbow spanning trees

An edge-colored graph is rainbow if no two edges have a same color. For two rainbow spanning trees \( T \) and \( T' \) of an edge-colored graph, we say that \( T' \) is obtained from \( T \) by an edge flip if there exist edges \( e \) and \( f \) in \( G \) such that \( T'=T-e+f \). For an edge-colored graph \( G \), we define the reconfiguration graph \( H \) to be the graph whose vertices correspond to rainbow spanning trees of \( G \), and rainbow spanning trees are joined by an edge in \( H \) if and only if one is obtained from the other by a single edge flip. The reconfiguration graph is not necessarily connected. We give a condition on an edge-colored graph for its reconfiguration graph to be connected. Moreover, we found some properties of reconfiguration graphs. I will talk about our results and some open problems. This is joint work with Yutaro Yamaguchi from Osaka University.

Stijn Cambie

Title: Winning strategies and graph colouring

We consider some problems of Erdős which are stated as games. Here two players alternatingly colour edges of a clique in their colour. At the end, they have coloured complementary graphs \( A \) and \( B \). For a graph parameter \( p \), depending on the relation between \( p (A) \) and \( p(B) \), one of them wins. In this talk, we will give examples of specific games, and ideas on the process.