Coloring directed cycles

Sopena in his survey [E. Sopena, The oriented chromatic number of graphs: A short survey, preprint 2013] writes, without any proof, that an oriented cycle $\vec C$ can be colored with three colors if and only if $\lambda(\vec C)=0$, where $\lambda(\vec C)$ is the number of forward arcs minus the number of backward arcs in $\vec C$. This is not true. In this paper we show that $\vec C$ can be colored with three colors if and only if $\lambda(\vec C)=0(\bmod~3)$ or $\vec C$ does not contain three consecutive arcs going in the same direction

Oriented Colouring Graphs of Bounded Degree and Degeneracy

This paper considers upper bounds on the oriented chromatic number, $\chi_o$, of graphs in terms of their maximum degree $\Delta$ and/or their degeneracy $d$. In particular we show that asymptotically, $\chi_o \leq \chi_2 f(d) 2^d$ where $f(d) \geq (\frac{1}{\log_2(e) -1} + \epsilon) d^2$ and $\chi_2 \leq 2^{\frac{f(d)}{d}}$. This improves a result of MacGillivray, Raspaud, and Swartz of the form $\chi_o \leq 2^{\chi_2} -1$. The rest of the paper is devoted to improving prior bounds for $\chi_o$ in terms of $\Delta$ and $d$ by refining the asymptotic arguments involved.Comment: 8 pages, 3 figure

Proper orientation of cacti

An orientation of a graph is proper if two adjacent vertices have different indegrees. We prove that every cactus admits a proper orientation with maximum indegree at most 7. We also prove that the bound 7 is tight by showing a cactus having no proper orientation with maximum indegree less than 7. We also prove that any planar claw-free graph has a proper orientation with maximum indegree at most 6 and that this bound can also be attained

On the Symmetries of Integrability

We show that the Yang-Baxter equations for two dimensional models admit as a group of symmetry the infinite discrete group $A_2^{(1)}$. The existence of this symmetry explains the presence of a spectral parameter in the solutions of the equations. We show that similarly, for three-dimensional vertex models and the associated tetrahedron equations, there also exists an infinite discrete group of symmetry. Although generalizing naturally the previous one, it is a much bigger hyperbolic Coxeter group. We indicate how this symmetry can help to resolve the Yang-Baxter equations and their higher-dimensional generalizations and initiate the study of three-dimensional vertex models. These symmetries are naturally represented as birational projective transformations. They may preserve non trivial algebraic varieties, and lead to proper parametrizations of the models, be they integrable or not. We mention the relation existing between spin models and the Bose-Messner algebras of algebraic combinatorics. Our results also yield the generalization of the condition $q^n=1$ so often mentioned in the theory of quantum groups, when no $q$ parameter is available.Comment: 23 page

Book embeddings of graphs

We use a structural theorem of Robertson and Seymour to show that for every minor-closed class of graphs, other than the class of all graphs, there is a number k such that every member of the class can be embedded in a book with k pages. Book embeddings of graphs with relation to surfaces, vertex extensions, clique-sums and r-rings are combined into a single book embedding of a graph in the minor-closed class. The effects of subdividing a complete graph and a complete bipartite graph with respect to book thickness are studied. We prove that if n ≥ 3, then the book thickness of Kn is the ceiling of (n/2). We also prove that for each m and B, there exists an integer N such that for all n ≥ ‪N, the book thickness of the graph obtained from subdividing each edge of Kn exactly m times has book thickness at least B. Additionally, there are corresponding theorems for complete bipartite graphs