Subgraphs, Closures and Hamiltonicity

Closure theorems in hamiltonian graph theory are of the following type: Let G be a 2- connected graph and let u, v be two distinct nonadjacent vertices of G. If condition c(u,v) holds, then G is hamiltonian if and only if G + uv is hamiltonian. We discuss several results of this type in which u and v are vertices of a subgraph H of G on four vertices and c(u, v) is a condition on the neighborhoods of the vertices of H (in G). We also discuss corresponding sufficient conditions for hamiltonicity of G

A closure concept based on neighborhood unions of independent triples

The well-known closure concept of Bondy and Chvatal is based on degree-sums of pairs of nonadjacent (independent) vertices. We show that a more general concept due to Ainouche and Christofides can be restated in terms of degree-sums of independent triples. We introduce a closure concept which is based on neighborhood unions of independent triples and which also generalizes the closure concept of Bondy and Chvatal

Homogeneous sets, clique-separators, critical graphs, and optimal χ\chi-binding functions

Given a set $\mathcal{H}$ of graphs, let $f_\mathcal{H}^\star\colon \mathbb{N}_{>0}\to \mathbb{N}_{>0}$ be the optimal $\chi$-binding function of the class of $\mathcal{H}$-free graphs, that is, $$f_\mathcal{H}^\star(\omega)=\max\{\chi(G): G\text{ is } \mathcal{H}\text{-free, } \omega(G)=\omega\}.$$ In this paper, we combine the two decomposition methods by homogeneous sets and clique-separators in order to determine optimal $\chi$-binding functions for subclasses of $P_5$-free graphs and of $(C_5,C_7,\ldots)$-free graphs. In particular, we prove the following for each $\omega\geq 1$: (i) $\ f_{\{P_5,banner\}}^\star(\omega)=f_{3K_1}^\star(\omega)\in \Theta(\omega^2/\log(\omega)),$ (ii) $\ f_{\{P_5,co-banner\}}^\star(\omega)=f^\star_{\{2K_2\}}(\omega)\in\mathcal{O}(\omega^2),$(iii)$\ f_{\{C_5,C_7,\ldots,banner\}}^\star(\omega)=f^\star_{\{C_5,3K_1\}}(\omega)\notin \mathcal{O}(\omega),$and (iv)$\ f_{\{P_5,C_4\}}^\star(\omega)=\lceil(5\omega-1)/4\rceil.$We also characterise, for each of our considered graph classes, all graphs$G$with$\chi(G)>\chi(G-u)$for each$u\in V(G)$. From these structural results, we can prove Reed's conjecture -- relating chromatic number, clique number, and maximum degree of a graph -- for$(P_5,banner)$-free graphs

Applications of Two Body Dirac Equations to Hadron and Positronium Spectroscopy

We review recent applications of the Two Body Dirac equations of constraint dynamics to meson spectroscopy and describe new extensions to three-body problems in their use in the study of baryon spectroscopy. We outline unique aspects of these equations for QED bound states that distinguish them among the various other approaches to the relativistic two body problem. Finally we discuss recent theorectial solutions of new peculiar bound states for positronium arising from the Two Body Dirac equations of constraint dynamics, assuming point particles for the electron and the positron.Comment: Invited talk: CST-MISC joint international symposium on particle physics - From spacetime dynamics to phenomenology - Tokyo, March 15-16, 201