The Incidence Chromatic Number of Toroidal Grids

An incidence in a graph $G$ is a pair $(v,e)$ with $v \in V(G)$ and $e \in E(G)$, such that $v$ and $e$ are incident. Two incidences $(v,e)$ and $(w,f)$ are adjacent if $v=w$, or $e=f$, or the edge $vw$ equals $e$ or $f$. The incidence chromatic number of $G$ is the smallest $k$ for which there exists a mapping from the set of incidences of $G$ to a set of $k$ colors that assigns distinct colors to adjacent incidences. In this paper, we prove that the incidence chromatic number of the toroidal grid $T_{m,n}=C_m\Box C_n$ equals 5 when $m,n \equiv 0 \pmod 5$ and 6 otherwise.Comment: 16 page

In Most 6-regular Toroidal Graphs All 5-colorings are Kempe Equivalent

A Kempe swap in a proper coloring interchanges the colors on some maximal connected 2-colored subgraph. Two $k$-colorings are $k$-equivalent if we can transform one into the other using Kempe swaps. The triangulated toroidal grid, $T[m\times n]$, is formed from (a toroidal embedding of) the Cartesian product of $C_m$ and $C_n$ by adding parallel diagonals inside all 4-faces. Mohar and Salas showed that not all 4-colorings of $T[m\times n]$ are 4-equivalent. In contrast, Bonamy, Bousquet, Feghali, and Johnson showed that all 6-colorings of $T[m\times n]$ are 6-equivalent. They asked whether the same is true for 5-colorings. We answer their question affirmatively when $m,n\ge 6$. Further, we show that if $G$ is 6-regular with a toroidal embedding where every non-contractible cycle has length at least 7, then all 5-colorings of $G$ are 5-equivalent. Our results relate to the antiferromagnetic Pott's model, in statistical mechanics.Comment: 17 pages, 16 figures; revised introduction; updated abstrac

Nucleon decay in gauge unified models with intersecting D6-branes

Baryon number violation is discussed in gauge unified orbifold models of type II string theory with intersecting Dirichlet branes. We consider setups of D6-branes which extend along the flat Minkowski space-time directions and wrap around 3-cycles of the internal 6-d manifold. The discussion is motivated by the enhancement effect of low energy amplitudes anticipated for M-theory and type II string theory models with matter modes localized at points of the internal manifold. The conformal field theory formalism is used to evaluate the open string amplitudes at tree level. We study the single baryon number violating processes of dimension 6 and 5, involving four quarks and leptons and in supersymmetry models, two pairs of matter fermions and superpartner sfermions. The higher order processes associated with the baryon number violating operators of dimension 7 and 9 are also examined, but in a qualitative way. We discuss the low energy representation of string theory amplitudes in terms of infinite series of poles associated to exchange of string Regge resonance and compactification modes. The comparison of string amplitudes with the equivalent field theory amplitudes is first studied in the large compactification radius limit. Proceeding next to the finite compactification radius case, we present a numerical study of the ratio of string to field theory amplitudes based on semi-realistic gauge unified non-supersymmetric and supersymmetric models employing the Z3 and Z2xZ2 orbifolds. We find a moderate enhancement of string amplitudes which becomes manifest in the regime where the gauge symmetry breaking mass parameter exceeds the compactification mass parameter, corresponding to a gauge unification in a seven dimensional space-time.Comment: 63 pages revtex4. 8 postscript figures. 4 tables. Subsection II.B revised. Several new references added. To appear in Physical Review