The chromatic numbers of double coverings of a graph

If we fix a spanning subgraph $H$ of a graph $G$, we can define a chromatic number of $H$ with respect to $G$ and we show that it coincides with the chromatic number of a double covering of $G$ with co-support $H$. We also find a few estimations for the chromatic numbers of $H$ with respect to $G$.Comment: 10 page

Cubical coloring -- fractional covering by cuts and semidefinite programming

We introduce a new graph invariant that measures fractional covering of a graph by cuts. Besides being interesting in its own right, it is useful for study of homomorphisms and tension-continuous mappings. We study the relations with chromatic number, bipartite density, and other graph parameters. We find the value of our parameter for a family of graphs based on hypercubes. These graphs play for our parameter the role that circular cliques play for the circular chromatic number. The fact that the defined parameter attains on these graphs the `correct' value suggests that the definition is a natural one. In the proof we use the eigenvalue bound for maximum cut and a recent result of Engstr\"om, F\"arnqvist, Jonsson, and Thapper. We also provide a polynomial time approximation algorithm based on semidefinite programming and in particular on vector chromatic number (defined by Karger, Motwani and Sudan [Approximate graph coloring by semidefinite programming, J. ACM 45 (1998), no. 2, 246--265]).Comment: 17 page

A Note on the Sparing Number of Graphs

An integer additive set-indexer is defined as an injective function $f:V(G)\rightarrow 2^{\mathbb{N}_0}$ such that the induced function $g_f:E(G) \rightarrow 2^{\mathbb{N}_0}$ defined by $g_f (uv) = f(u)+ f(v)$ is also injective. An IASI $f$ is said to be a weak IASI if $|g_f(uv)|=max(|f(u)|,|f(v)|)$ for all $u,v\in V(G)$. A graph which admits a weak IASI may be called a weak IASI graph. The set-indexing number of an element of a graph $G$, a vertex or an edge, is the cardinality of its set-labels. The sparing number of a graph $G$ is the minimum number of edges with singleton set-labels, required for a graph $G$ to admit a weak IASI. In this paper, we study the sparing number of certain graphs and the relation of sparing number with some other parameters like matching number, chromatic number, covering number, independence number etc.Comment: 10 pages, 10 figures, submitte

On asymptotic packing of convex geometric and ordered graphs

A convex geometric graph $G$ is said to be packable if there exist edge-disjoint copies of $G$ in the complete convex geometric graph $K_n$ covering all but $o(n^2)$ edges. We prove that every convex geometric graph with cyclic chromatic number at most $4$ is packable. With a similar definition of packability for ordered graphs, we prove that every ordered graph with interval chromatic number at most $3$ is packable. Arguments based on the average length of edges imply these results are best possible. We also identify a class of convex geometric graphs that are packable due to having many "long" edges

An asymptotic multipartite Kühn-Osthus theorem

In this paper we prove an asymptotic multipartite version of a well-known theorem of K¨uhn and Osthus by establishing, for any graph H with chromatic number r, the asymptotic multipartite minimum degree threshold which ensures that a large r-partite graph G admits a perfect H-tiling. We also give the threshold for an H-tiling covering all but a linear number of vertices of G, in a multipartite analogue of results of Koml´os and of Shokoufandeh and Zhao

Classifying Derived Voltage Graphs

Gross and Tucker’s voltage graph construction assigns group elements as weights to the edges of an oriented graph. This construction provides a blueprint for inducing graph covers. Thomas Zaslavsky studies the criteria for balance in voltage graphs. This project primarily examines the relationship between the group structure of the set of all possible assignments of a group to a graph, including the balanced subgroup, and the isomorphism classes of covering graphs. We examine connectedness, planarity, and chromatic number in the derived graph. Lastly we explain the future research possibilities involving the fundamental group