Random polymers on the complete graph

Consider directed polymers in a random environment on the complete graph of size $N$. This model can be formulated as a product of i.i.d. $N\times N$ random matrices and its large time asymptotics is captured by Lyapunov exponents and the Furstenberg measure. We detail this correspondence, derive the long-time limit of the model and obtain a co-variant distribution for the polymer path. Next, we observe that the model becomes exactly solvable when the disorder variables are located on edges of the complete graph and follow a totally asymmetric stable law of index $\alpha \in (0,1)$. Then, a certain notion of mean height of the polymer behaves like a random walk and we show that the height function is distributed around this mean according to an explicit law. Large $N$ asymptotics can be taken in this setting, for instance, for the free energy of the system and for the invariant law of the polymer height with a shift. Moreover, we give some perturbative results for environments which are close to the totally asymmetric stable laws

Complete graph immersions and minimum degree

An immersion of a graph H in another graph G is a one-to-one mapping phi:V(H)->V(G) and a collection of edge-disjoint paths in G, one for each edge of H, such that the path P_{uv} corresponding to the edge uv has endpoints phi(u) and phi(v). The immersion is strong if the paths P_{uv} are internally disjoint from phi(V(H)). We prove that every simple graph of minimum degree at least 11t+7 contains a strong immersion of the complete graph K_t. This improves on previously known bound of minimum degree at least 200t obtained by DeVos et al.Comment: 12 pages, 1 figur

Complete graph immersions in dense graphs

In this article we consider the relationship between vertex coloring and the immersion order. Specifically, a conjecture proposed by Abu-Khzam and Langston in 2003, which says that the complete graph with $t$ vertices can be immersed in any $t$-chromatic graph, is studied. First, we present a general result about immersions and prove that the conjecture holds for graphs whose complement does not contain any induced cycle of length four and also for graphs having the property that every set of five vertices induces a subgraph with at least six edges. Then, we study the class of all graphs with independence number less than three, which are graphs of interest for Hadwiger's Conjecture. We study such graphs for the immersion-analog. If Abu-Khzam and Langston's conjecture is true for this class of graphs, then an easy argument shows that every graph of independence number less than $3$ contains $K_{\left\lceil\frac{n}{2}\right\rceil}$ as an immersion. We show that the converse is also true. That is, if every graph with independence number less than $3$ contains an immersion of $K_{\left\lceil\frac{n}{2}\right\rceil}$, then Abu-Khzam and Langston's conjecture is true for this class of graphs. Furthermore, we show that every graph of independence number less than $3$ has an immersion of $K_{\left\lceil\frac{n}{3}\right\rceil}$

Complete graph decompositions and p-groupoids

We study P-groupoids that arise from certain decompositions of complete graphs. We show that left distributive P-groupoids are distributive, quasigroups. We characterize P-groupoids when the corresponding decomposition is a Hamiltonian decomposition for complete graphs of odd, prime order. We also study a specific example of a P-quasigroup constructed from cyclic groups of odd order. We show such P-quasigroups have characteristic left and right multiplication groups, as well as the right multiplication group is isomorphic to the dihedral group

Face distributions of embeddings of complete graphs

A longstanding open question of Archdeacon and Craft asks whether every complete graph has a minimum genus embedding with at most one nontriangular face. We exhibit such an embedding for each complete graph except $K_8$, the complete graph on 8 vertices, and we go on to prove that no such embedding can exist for this graph. Our approach also solves a more general problem, giving a complete characterization of the possible face distributions (i.e. the numbers of faces of each length) realizable by minimum genus embeddings of each complete graph. We also tackle analogous questions for nonorientable and maximum genus embeddings.Comment: 37 pages, 32 figure

The Naming Game on the complete graph

We consider a model of language development, known as the naming game, in which agents invent, share and then select descriptive words for a single object, in such a way as to promote local consensus. When formulated on a finite and connected graph, a global consensus eventually emerges in which all agents use a common unique word. Previous numerical studies of the model on the complete graph with $n$ agents suggest that when no words initially exist, the time to consensus is of order $n^{1/2}$, assuming each agent speaks at a constant rate. We show rigorously that the time to consensus is at least $n^{1/2-o(1)}$, and that it is at most constant times $\log n$ when only two words remain. In order to do so we develop sample path estimates for quasi-left continuous semimartingales with bounded jumps.Comment: 34 pages, no figure

Self-avoiding walk on the complete graph

There is an extensive literature concerning self-avoiding walk on infinite graphs, but the subject is relatively undeveloped on finite graphs. The purpose of this paper is to elucidate the phase transition for self-avoiding walk on the simplest finite graph: the complete graph. We make the elementary observation that the susceptibility of the self-avoiding walk on the complete graph is given exactly in terms of the incomplete gamma function. The known asymptotic behaviour of the incomplete gamma function then yields a complete description of the finite-size scaling of the self-avoiding walk on the complete graph. As a basic example, we compute the limiting distribution of the length of a self-avoiding walk on the complete graph, in subcritical, critical, and supercritical regimes. This provides a prototype for more complex unsolved problems such as the self-avoiding walk on the hypercube or on a high-dimensional torus.Comment: 11 pages. Minor edits, to be published in J. Math. Soc. Japa

The bunkbed conjecture on the complete graph

The bunkbed conjecture was first posed by Kasteleyn. If $G=(V,E)$ is a finite graph and $H$ some subset of $V$, then the bunkbed of the pair $(G,H)$ is the graph $G\times\{1,2\}$ plus $|H|$ extra edges to connect for every $v\in H$ the vertices $(v,1)$ and $(v,2)$. The conjecture asserts that $(v,1)$ is more likely to connect with $(w,1)$ than with $(w,2)$ in the independent bond percolation model for any $v,w\in V$. This is intuitive because $(v,1)$ is in some sense closer to $(w,1)$ than it is to $(w,2)$. The conjecture has however resisted several attempts of proof. This paper settles the conjecture in the case of a constant percolation parameter and $G$ the complete graph.Comment: 3 pages; published in the European Journal of Combinatorics: see https://doi.org/10.1016/j.ejc.2018.10.00

Frog model wakeup time on the complete graph

The frog model is a system of random walks where active particles set sleeping particles in motion. On the complete graph with n vertices it is equivalent to a well-understood rumor spreading model. We given an alternate and elementary proof that the wake-up time, i.e. the expected time for every particle to be activated, is Theta(log n). Additionally, we give an explicit distributional equation for the wakeup time as a weighted sum of geometric random variables. This project was part of the University of Washington Research Experience for Undergraduates program.Comment: 9 page

An Average Case NP-Complete Graph Coloring Problem

NP-complete problems should be hard on some instances but those may be extremely rare. On generic instances many such problems, especially related to random graphs, have been proven easy. We show the intractability of random instances of a graph coloring problem: this graph problem is hard on average unless all NP problem under all samplable (i.e., generatable in polynomial time) distributions are easy. Worst case reductions use special gadgets and typically map instances into a negligible fraction of possible outputs. Ours must output nearly random graphs and avoid any super-polynomial distortion of probabilities.Comment: 15 page