arXiv subjects

Bojan Mohar

Publications and source records attributed to Bojan Mohar.

149 records · Page 9Linked to original sources

arXiv2009

A new Kempe invariant and the (non)-ergodicity of the Wang-Swendsen-Kotecky algorithm

We prove that for the class of three-colorable triangulations of a closed oriented surface, the degree of a four-coloring modulo 12 is an invariant under Kempe changes. We use this general result to prove that for all triangulations T(3L,3M) of the torus with 3<= L <= M, there are at least two Kempe equivalence classes. This result implies in particular that the Wang-Swendsen-Kotecky algorithm for the zero-temperature 4-state Potts antiferromagnet on these triangulations T(3L,3M) of the torus is not ergodic.

math.CO↗

arXiv2009

Simplices and spectra of graphs, continued

In this note we show that the n-2-dimensional volumes of codimension 2 faces of an n-dimensional simplex are algebraically independent quantities of the volumes of its edge-lengths. The proof involves computation of the eigenvalues of Kneser graphs. We also construct families of non-congurent simplices not determined by their codimension-2 areas.

math.MG↗

arXiv2008

Degenerate and star colorings of graphs on surfaces

We study the degenerate, the star and the degenerate star chromatic numbers and their relation to the genus of graphs. As a tool we prove the following strengthening of a result of Fertin et al.: If $G$ is a graph of maximum degree $Δ$, then $G$ admits a degenerate star coloring using $O(Δ^{3/2})$ colors. We use this result to prove that every graph of genus $g$ admits a degenerate star coloring with $O(g^{3/5})$ colors. It is also shown that these results are sharp up to a logarithmic factor.

math.CO↗

arXiv2007

Cayley sum graphs and eigenvalues of $(3,6)$-fullerenes

We determine the spectra of cubic plane graphs whose faces have sizes 3 and 6. Such graphs, "(3,6)-fullerenes", have been studied by chemists who are interested in their energy spectra. In particular we prove a conjecture of Fowler, which asserts that all their eigenvalues come in pairs of the form $\{λ,-λ\}$ except for the four eigenvalues $\{3,-1,-1,-1\}$. We exhibit other families of graphs which are "spectrally nearly bipartite" in this sense. Our proof utilizes a geometric representation to recognize the algebraic structure of these graphs, which turn out to be examples of Cayley sum graphs.

math.CO↗

arXiv2007

A quadratic lower bound for subset sums

Let A be a finite nonempty subset of an additive abelian group G, and let Σ(A) denote the set of all group elements representable as a sum of some subset of A. We prove that |Σ(A)| >= |H| + 1/64 |A H|^2 where H is the stabilizer of Σ(A). Our result implies that Σ(A) = Z/nZ for every set A of units of Z/nZ with |A| >= 8 \sqrt{n}. This consequence was first proved by Erdős and Heilbronn for n prime, and by Vu (with a weaker constant) for general n.

math.NT↗