Cycles through three vertices in 2-connected graphs

Let Gbe a 2-connected graph with minimum degree d and let {x, y, z} be a set of three vertices contained on some cycle ofG. ThenG ishamiltonian or {x, y, z} is contained on a cycle of length at least 2d inG.     

Nowhere-Zero 4-Flows,Simultaneous Edge-Colorings,And Critical Partial Latin Squares

It is proved in this paper that every bipartite graphic sequence with the minimum degree 2 has a realization that admits a nowhere-zero 4-flow. This result implies a conjecture originally proposed by Keedwell (1993) and reproposed by Cameron (1999) about simultaneous edge-colorings and critical partial Latin squares.* Partially supported by RGC grant HKU7054/03P. Partially supported by the National Security Agency under Grants MDA904-00-1-00614 and MDA904-01-1-0022.     

On the Length of Snakes in Powers of Complete Graphs

The conjecture stated in an earlier paper by the author thatthere is a constant (independent from both n and k) such that n^d–1 holds for everyn 2 and d 2, where is the length of the longest snake (cycle without chords) in the Cartesianproduct of d copies of thecomplete graph K_n, is proved.     

Every 4-Connected Line Graph of a Planar Graph is Hamiltonian

Let G be a graph withE(G) $#x2260;ø. The line graph of G, written L(G) hasE(G) as its vertex set, where two vertices are adjacent in L(G) if and only if the corresponding edges are adjacent inG. Thomassen conjectured that all 4-connected line graphs are hamiltonian [2]. We show that this conjecture holds for planar graphs.     

Matchings Including or Excluding Certain Edge Sets in Bipartite Graphs

Necessary and sufficient conditions are obtained for a bipartitegraph to admit a matching which includes a preassigned set ofedges and excludes another preassigned set of edges. 1991 MathematicsSubject Classification 05C70.     

The optimization of the domain problem. I. Basic concepts

We introduce a representation of geometric shapes by vectors in an inner product space. A suitable renormalization procedure results in bounded regions of $\text{R}$ⁿ being represented by vectors of finite length.     

Chemical reaction-diffusion networks: convergence of the method of lines

We show that solutions of the chemical reaction-diffusion system associated to \(A+B\rightleftharpoons C\) in one spatial dimension can be approximated in \(L^2\) on any finite time interval by solutions of a space discretized ODE system which models the corresponding chemical reaction system replicated in the discretization subdomains where the concentrations are assumed spatially constant. Same-species reactions through the virtual boundaries of adjacent subdomains lead to diffusion in the vanishing limit. We show convergence of our numerical scheme by way of a consistency estimate, with features generalizable to reaction networks other than the one considered here, and to multiple space dimensions. In particular, the connection with the class of complex-balanced systems is briefly discussed here, and will be considered in future work.