Doubly Chorded Cycles in Graphs

In 1963, Corradi and Hajnal proved that for any positive integer k if a graph contains at least 3k vertices and has minimum degree at least 2k, then it contains k disjoint cycles. This result is sharp, meaning there are graphs on at least 3k vertices with a minimum degree of 2k-1 that do not contain k disjoint cycles. Their work is the motivation behind finding sharp conditions that guarantee the existence of specific structures, e.g. cycles, chorded cycles, theta graphs, etc. In this talk, we will explore minimum degree conditions which guarantee a specific number of doubly chorded cycles, graphs that contain a spanning cycle and at least two additional edges, called chords. In particular, we will discuss our findings on these conditions and how it fits in with previous results in this area

A general theorem in spectral extremal graph theory

The extremal graphs $\mathrm{EX}(n,\mathcal F)$ and spectral extremal graphs $\mathrm{SPEX}(n,\mathcal F)$ are the sets of graphs on $n$ vertices with maximum number of edges and maximum spectral radius, respectively, with no subgraph in $\mathcal F$. We prove a general theorem which allows us to characterize the spectral extremal graphs for a wide range of forbidden families $\mathcal F$ and implies several new and existing results. In particular, whenever $\mathrm{EX}(n,\mathcal F)$ contains the complete bipartite graph $K_{k,n-k}$ (or certain similar graphs) then $\mathrm{SPEX}(n,\mathcal F)$ contains the same graph when $n$ is sufficiently large. We prove a similar theorem which relates $\mathrm{SPEX}(n,\mathcal F)$ and $\mathrm{SPEX}_\alpha(n,\mathcal F)$, the set of $\mathcal F$-free graphs which maximize the spectral radius of the matrix $A_\alpha=\alpha D+(1-\alpha)A$, where $A$ is the adjacency matrix and $D$ is the diagonal degree matrix

β-Perfect Graphs

AbstractThe class ofβ-perfect graphs is introduced. We draw a number of parallels between these graphs and perfect graphs. We also introduce some special classes ofβ-perfect graphs. Finally, we show that the greedy algorithm can be used to colour a graphGwith no even chordless cycles using at most 2(χ(G)−1) colours