On the L(p,1)L(p,1)-labelling of graphs

In this paper we improve the best known bound for the $L(p,1)$-labelling of graphs with given maximal degree

Chromatic Polynomials for J(∏H)IJ(\prod H)I Strip Graphs and their Asymptotic Limits

We calculate the chromatic polynomials $P$ for $n$-vertex strip graphs of the form $J(\prod_{\ell=1}^m H)I$, where $J$ and $I$ are various subgraphs on the left and right ends of the strip, whose bulk is comprised of $m$-fold repetitions of a subgraph $H$. The strips have free boundary conditions in the longitudinal direction and free or periodic boundary conditions in the transverse direction. This extends our earlier calculations for strip graphs of the form $(\prod_{\ell=1}^m H)I$. We use a generating function method. From these results we compute the asymptotic limiting function $W=\lim_{n \to \infty}P^{1/n}$; for $q \in {\mathbb Z}_+$ this has physical significance as the ground state degeneracy per site (exponent of the ground state entropy) of the $q$-state Potts antiferromagnet on the given strip. In the complex $q$ plane, $W$ is an analytic function except on a certain continuous locus ${\cal B}$. In contrast to the $(\prod_{\ell=1}^m H)I$ strip graphs, where ${\cal B}$ (i) is independent of $I$, and (ii) consists of arcs and possible line segments that do not enclose any regions in the $q$ plane, we find that for some $J(\prod_{\ell=1}^m H)I$ strip graphs, ${\cal B}$ (i) does depend on $I$ and $J$, and (ii) can enclose regions in the $q$ plane. Our study elucidates the effects of different end subgraphs $I$ and $J$ and of boundary conditions on the infinite-length limit of the strip graphs.Comment: 33 pages, Latex, 7 encapsulated postscript figures, Physica A, in press, with some typos fixe

The complexity of the L(p,q)-labeling problem for bipartite planar graphs of small degree

AbstractGiven a simple graph G, by an L(p,q)-labeling of G we mean a function c that assigns nonnegative integers to its vertices in such a way that if two vertices u, v are adjacent then |c(u)−c(v)|≥p, and if they are at distance 2 then |c(u)−c(v)|≥q. The L(p,q)-labeling problem can be defined as follows: given a graph G and integer t, determine whether there exists an L(p,q)-labeling c of G such that c(V)⊆{0,1,…,t}. In the paper we show that the problem is NP-complete even when restricted to bipartite planar graphs of small maximum degree and for relatively small values of t. More precisely, we prove that: (1)if p<3q then the problem is NP-complete for bipartite planar graphs of maximum degree Δ≤3 and t=p+max{2q,p};(2)if p=3q then the problem is NP-complete for bipartite planar graphs of maximum degree Δ≤4 and t=6q;(3)if p>3q then the problem is NP-complete for bipartite planar graphs of maximum degree Δ≤4 and t=p+5q.In particular, these results imply that the L(2,1)-labeling problem in planar graphs is NP-complete for t=4, and that the L(p,q)-labeling problem in graphs of maximum degree Δ≤4 is NP-complete for all values of p and q, thus answering two well-known open questions

Characterization and enumeration of toroidal K_{3,3}-subdivision-free graphs

We describe the structure of 2-connected non-planar toroidal graphs with no K_{3,3}-subdivisions, using an appropriate substitution of planar networks into the edges of certain graphs called toroidal cores. The structural result is based on a refinement of the algorithmic results for graphs containing a fixed K_5-subdivision in [A. Gagarin and W. Kocay, "Embedding graphs containing K_5-subdivisions'', Ars Combin. 64 (2002), 33-49]. It allows to recognize these graphs in linear-time and makes possible to enumerate labelled 2-connected toroidal graphs containing no K_{3,3}-subdivisions and having minimum vertex degree two or three by using an approach similar to [A. Gagarin, G. Labelle, and P. Leroux, "Counting labelled projective-planar graphs without a K_{3,3}-subdivision", submitted, arXiv:math.CO/0406140, (2004)].Comment: 18 pages, 7 figures and 4 table

On prisms, M\"obius ladders and the cycle space of dense graphs

For a graph X, let f_0(X) denote its number of vertices, d(X) its minimum degree and Z_1(X;Z/2) its cycle space in the standard graph-theoretical sense (i.e. 1-dimensional cycle group in the sense of simplicial homology theory with Z/2-coefficients). Call a graph Hamilton-generated if and only if the set of all Hamilton circuits is a Z/2-generating system for Z_1(X;Z/2). The main purpose of this paper is to prove the following: for every s > 0 there exists n_0 such that for every graph X with f_0(X) >= n_0 vertices, (1) if d(X) >= (1/2 + s) f_0(X) and f_0(X) is odd, then X is Hamilton-generated, (2) if d(X) >= (1/2 + s) f_0(X) and f_0(X) is even, then the set of all Hamilton circuits of X generates a codimension-one subspace of Z_1(X;Z/2), and the set of all circuits of X having length either f_0(X)-1 or f_0(X) generates all of Z_1(X;Z/2), (3) if d(X) >= (1/4 + s) f_0(X) and X is square bipartite, then X is Hamilton-generated. All these degree-conditions are essentially best-possible. The implications in (1) and (2) give an asymptotic affirmative answer to a special case of an open conjecture which according to [European J. Combin. 4 (1983), no. 3, p. 246] originates with A. Bondy.Comment: 33 pages; 5 figure

Asymptotic enumeration and limit laws for graphs of fixed genus

It is shown that the number of labelled graphs with n vertices that can be embedded in the orientable surface S_g of genus g grows asymptotically like $c^{(g)}n^{5(g-1)/2-1}\gamma^n n!$ where $c^{(g)}>0$, and $\gamma \approx 27.23$ is the exponential growth rate of planar graphs. This generalizes the result for the planar case g=0, obtained by Gimenez and Noy. An analogous result for non-orientable surfaces is obtained. In addition, it is proved that several parameters of interest behave asymptotically as in the planar case. It follows, in particular, that a random graph embeddable in S_g has a unique 2-connected component of linear size with high probability