Distance colouring without one cycle length

We consider distance colourings in graphs of maximum degree at most $d$ and how excluding one fixed cycle length $\ell$ affects the number of colours required as $d\to\infty$. For vertex-colouring and $t\ge 1$, if any two distinct vertices connected by a path of at most $t$ edges are required to be coloured differently, then a reduction by a logarithmic (in $d$) factor against the trivial bound $O(d^t)$ can be obtained by excluding an odd cycle length $\ell \ge 3t$ if $t$ is odd or by excluding an even cycle length $\ell \ge 2t+2$. For edge-colouring and $t\ge 2$, if any two distinct edges connected by a path of fewer than $t$ edges are required to be coloured differently, then excluding an even cycle length $\ell \ge 2t$ is sufficient for a logarithmic factor reduction. For $t\ge 2$, neither of the above statements are possible for other parity combinations of $\ell$ and $t$. These results can be considered extensions of results due to Johansson (1996) and Mahdian (2000), and are related to open problems of Alon and Mohar (2002) and Kaiser and Kang (2014).Comment: 14 pages, 1 figur

Sabidussi Versus Hedetniemi for Three Variations of the Chromatic Number

We investigate vector chromatic number, Lovasz theta of the complement, and quantum chromatic number from the perspective of graph homomorphisms. We prove an analog of Sabidussi's theorem for each of these parameters, i.e. that for each of the parameters, the value on the Cartesian product of graphs is equal to the maximum of the values on the factors. We also prove an analog of Hedetniemi's conjecture for Lovasz theta of the complement, i.e. that its value on the categorical product of graphs is equal to the minimum of its values on the factors. We conjecture that the analogous results hold for vector and quantum chromatic number, and we prove that this is the case for some special classes of graphs.Comment: 18 page

Topological lower bounds for the chromatic number: A hierarchy

This paper is a study of ``topological'' lower bounds for the chromatic number of a graph. Such a lower bound was first introduced by Lov\'asz in 1978, in his famous proof of the \emph{Kneser conjecture} via Algebraic Topology. This conjecture stated that the \emph{Kneser graph} \KG_{m,n}, the graph with all $k$-element subsets of $\{1,2,...,n\}$ as vertices and all pairs of disjoint sets as edges, has chromatic number $n-2k+2$. Several other proofs have since been published (by B\'ar\'any, Schrijver, Dolnikov, Sarkaria, Kriz, Greene, and others), all of them based on some version of the Borsuk--Ulam theorem, but otherwise quite different. Each can be extended to yield some lower bound on the chromatic number of an arbitrary graph. (Indeed, we observe that \emph{every} finite graph may be represented as a generalized Kneser graph, to which the above bounds apply.) We show that these bounds are almost linearly ordered by strength, the strongest one being essentially Lov\'asz' original bound in terms of a neighborhood complex. We also present and compare various definitions of a \emph{box complex} of a graph (developing ideas of Alon, Frankl, and Lov\'asz and of \kriz). A suitable box complex is equivalent to Lov\'asz' complex, but the construction is simpler and functorial, mapping graphs with homomorphisms to $\Z_2$-spaces with $\Z_2$-maps.Comment: 16 pages, 1 figure. Jahresbericht der DMV, to appea

A new property of the Lov\'asz number and duality relations between graph parameters

We show that for any graph $G$, by considering "activation" through the strong product with another graph $H$, the relation $\alpha(G) \leq \vartheta(G)$ between the independence number and the Lov\'{a}sz number of $G$ can be made arbitrarily tight: Precisely, the inequality $$\alpha(G \times H) \leq \vartheta(G \times H) = \vartheta(G)\,\vartheta(H)$$ becomes asymptotically an equality for a suitable sequence of ancillary graphs $H$. This motivates us to look for other products of graph parameters of $G$ and $H$ on the right hand side of the above relation. For instance, a result of Rosenfeld and Hales states that $$\alpha(G \times H) \leq \alpha^*(G)\,\alpha(H),$$ with the fractional packing number $\alpha^*(G)$, and for every $G$ there exists $H$ that makes the above an equality; conversely, for every graph $H$ there is a $G$ that attains equality. These findings constitute some sort of duality of graph parameters, mediated through the independence number, under which $\alpha$ and $\alpha^*$ are dual to each other, and the Lov\'{a}sz number $\vartheta$ is self-dual. We also show duality of Schrijver's and Szegedy's variants $\vartheta^-$ and $\vartheta^+$ of the Lov\'{a}sz number, and explore analogous notions for the chromatic number under strong and disjunctive graph products.Comment: 16 pages, submitted to Discrete Applied Mathematics for a special issue in memory of Levon Khachatrian; v2 has a full proof of the duality between theta+ and theta- and a new author, some new references, and we corrected several small errors and typo

Topological obstructions for vertex numbers of Minkowski sums

We show that for polytopes P_1, P_2, ..., P_r \subset \R^d, each having n_i \ge d+1 vertices, the Minkowski sum P_1 + P_2 + ... + P_r cannot achieve the maximum of \prod_i n_i vertices if r \ge d. This complements a recent result of Fukuda & Weibel (2006), who show that this is possible for up to d-1 summands. The result is obtained by combining methods from discrete geometry (Gale transforms) and topological combinatorics (van Kampen--type obstructions) as developed in R\"{o}rig, Sanyal, and Ziegler (2007).Comment: 13 pages, 2 figures; Improved exposition and less typos. Construction/example and remarks adde

Bounds on the Complex Zeros of (Di)Chromatic Polynomials and Potts-Model Partition Functions

I show that there exist universal constants $C(r) < \infty$ such that, for all loopless graphs $G$ of maximum degree $\le r$, the zeros (real or complex) of the chromatic polynomial $P_G(q)$ lie in the disc $|q| < C(r)$. Furthermore, $C(r) \le 7.963906... r$. This result is a corollary of a more general result on the zeros of the Potts-model partition function $Z_G(q, {v_e})$ in the complex antiferromagnetic regime $|1 + v_e| \le 1$. The proof is based on a transformation of the Whitney-Tutte-Fortuin-Kasteleyn representation of $Z_G(q, {v_e})$ to a polymer gas, followed by verification of the Dobrushin-Koteck\'y-Preiss condition for nonvanishing of a polymer-model partition function. I also show that, for all loopless graphs $G$ of second-largest degree $\le r$, the zeros of $P_G(q)$ lie in the disc $|q| < C(r) + 1$. Along the way, I give a simple proof of a generalized (multivariate) Brown-Colbourn conjecture on the zeros of the reliability polynomial for the special case of series-parallel graphs.Comment: 47 pages (LaTeX). Revised version contains slightly simplified proofs of Propositions 4.2 and 4.5. Version 3 fixes a silly error in my proof of Proposition 4.1, and adds related discussion. To appear in Combinatorics, Probability & Computin