On cuts of small chromatic number in sparse graphs

We say a graph has property $\mathcal{P}_{g,p}$ when it is an induced subgraph of the curve graph of a surface of genus $g$ with $p$ punctures. Two well-known graph invariants, the chromatic and clique numbers, can provide obstructions to $\mathcal{P}_{g,p}$. We introduce a new invariant of a graph, the \emph{nested complexity length}, which provides a novel obstruction to $\mathcal{P}_{g,p}$. For the curve graph this invariant captures the topological complexity of the surface in graph-theoretic terms; indeed we show that its value is $6g-6+2p$, i.e. twice the size of a maximal multicurve on the surface. As a consequence we show that large `half-graphs' do not have $\mathcal{P}_{g,p}$, and we deduce quantitatively that almost all finite graphs which pass the chromatic and clique tests do not have $\mathcal{P}_{g,p}$. We also reinterpret our obstruction in terms of the first-order theory of the curve graph, and in terms of RAAG subgroups of the mapping class group (following Kim and Koberda). Finally, we show that large complete multipartite graphs cannot have $\mathcal{P}_{g,p}$. This allows us to compute the upper density of the curve graph, and to conclude that clique size, chromatic number, and nested complexity length are not sufficient to determine $\mathcal{P}_{g,p}$.

Homomorphisms from sparse graphs with large girth

A new graph coloring algorithm is presented and compared to a wide variety of known algorithms. The algorithm is shown to exhibit O(n 2) time behavior for most sparse graphs and thus is found to be particularly well suited for use with large-scale scheduling problems. In addition, a procedure for generating large random test graphs with known chromatic number is presented and is used to evaluate heuristically the capabilities of the algorithms discussed.

Let C(G) denote the set of lengths of cycles in a graph G. In the first part of this paper, we study the minimum possible value of |C(G)| over all graphs G of average degree d and girth g. Erdős [8] conjectured that |C(G)|=Ω ` d (g−1)/2 for all such graphs, and we prove this conjecture. In particular, the longest cycle in a graph of average degree d and girth g has length Ω ` d (g−1)/2 . The study of this problem was initiated by Ore in 1967 and our result improves all previously known lower bounds on the length of the longest cycle [7,11,21,24,25]. Moreover, our bound cannot be improved in general, since known constructions of d-regular Moore Graphs of girth g have roughly that many vertices. We also show that Ω ` d (g−1)/2 is a lower bound for the number of odd cycle lengths in a graph of chromatic number d and girth g. Further results are obtained for the number of cycle lengths in H-free graphs of average degree d.

Kim and Vu made the following conjecture (Advances in Mathematics, 2004): if dgg log n, then the random d-regular graph {mathscr {G}}(n,d) can asymptotically almost surely be “sandwiched” between {mathscr {G}}(n,p_1) and {mathscr {G}}(n,p_2) where p_1 and p_2 are both (1+o(1))d/n. They proved this conjecture for log nll dleqslant n^{1/3-o(1)}, with a defect in the sandwiching: {mathscr {G}}(n,d) contains {mathscr {G}}(n,p_1) perfectly, but is not completely contained in {mathscr {G}}(n,p_2). The embedding {mathscr {G}}(n,p_1) subseteq {mathscr {G}}(n,d) was improved by Dudek, Frieze, Ruciński and Šileikis to d=o(n). In this paper, we prove Kim–Vu’s sandwich conjecture, with perfect containment on both sides, for all d where min {d, n-d}gg n/sqrt{log n}. The sandwich theorem allows translation of many results from {mathscr {G}}(n,p) to {mathscr {G}}(n,d) such as Hamiltonicity, the chromatic number, the diameter, etc. It also allows translation of threshold functions of phase transitions from {mathscr {G}}(n,p) to bond percolation of {mathscr {G}}(n,d). In addition to sandwiching regular graphs, our results cover graphs whose degrees are asymptotically equal. The proofs rely on estimates for the probability of small subgraph appearances in a random factor of a pseudorandom graph, which is of independent interest.

Given a sequence of nonnegative real numbers λ0, λ1… which sum to 1, we consider random graphs having approximately λi n vertices of degree i. Essentially, we show that if Σ i(i ‐ 2)λi > 0, then such graphs almost surely have a giant component, while if Σ i(i ‐2)λ. < 0, then almost surely all components in such graphs are small. We can apply these results to Gn,p,Gn.M, and other well‐known models of random graphs. There are also applications related to the chromatic number of sparse random graphs.

We investigate the linear chromatic number χlin(G(n,p)) of the binomial random graph G(n,p) on n vertices in which each edge appears independently with probability p=p(n). For a graph G, χlin(G) is defined as the smallest k such that G admits a k-colouring with the property that every path P in G receives a colour which appears on only one vertex of P. For dense random graphs (np→∞ as n→∞), we show that asymptotically almost surely χlin(G(n,p))≥n(1−O((np)−1/2))=n(1−o(1)). Understanding the order of the linear chromatic number for subcritical random graphs (np<1) and critical ones (np=1) is relatively easy. However, supercritical sparse random graphs (np=c for some constant c>1) remain to be investigated.

A note on the chromatic number of a dense random graph

Proper conflict-free list-coloring, odd minors, subdivisions, and layered treewidth