On the classification problem for split graphs

Abstract The Classification Problem is the problem of deciding whether a simple graph has chromatic index equal to Δ or Δ+1. In the first case, the graphs are called Class 1, otherwise, they are Class 2. A split graph is a graph whose vertex set admits a partition into a stable set and a clique. Split graphs are a subclass of chordal graphs. Figueiredo at al. (J. Combin. Math. Combin. Comput. 32:79–91, 2000) state that a chordal graph is Class 2 if and only if it is neighborhood-overfull. In this paper, we give a characterization of neighborhood-overfull split graphs and we show that the above conjecture is true for some split graphs

Bandwidth theorem for random graphs

A graph $G$ is said to have \textit{bandwidth} at most $b$, if there exists a labeling of the vertices by $1,2,..., n$, so that $|i - j| \leq b$ whenever $\{i,j\}$ is an edge of $G$. Recently, B\"{o}ttcher, Schacht, and Taraz verified a conjecture of Bollob\'{a}s and Koml\'{o}s which says that for every positive $r,\Delta,\gamma$, there exists $\beta$ such that if $H$ is an $n$-vertex $r$-chromatic graph with maximum degree at most $\Delta$ which has bandwidth at most $\beta n$, then any graph $G$ on $n$ vertices with minimum degree at least $(1 - 1/r + \gamma)n$ contains a copy of $H$ for large enough $n$. In this paper, we extend this theorem to dense random graphs. For bipartite $H$, this answers an open question of B\"{o}ttcher, Kohayakawa, and Taraz. It appears that for non-bipartite $H$ the direct extension is not possible, and one needs in addition that some vertices of $H$ have independent neighborhoods. We also obtain an asymptotically tight bound for the maximum number of vertex disjoint copies of a fixed $r$-chromatic graph $H_0$ which one can find in a spanning subgraph of $G(n,p)$ with minimum degree $(1-1/r + \gamma)np$.Comment: 29 pages, 3 figure

Chromatic roots are dense in the whole complex plane

I show that the zeros of the chromatic polynomials P_G(q) for the generalized theta graphs \Theta^{(s,p)} are, taken together, dense in the whole complex plane with the possible exception of the disc |q-1| < 1. The same holds for their dichromatic polynomials (alias Tutte polynomials, alias Potts-model partition functions) Z_G(q,v) outside the disc |q+v| < |v|. An immediate corollary is that the chromatic zeros of not-necessarily-planar graphs are dense in the whole complex plane. The main technical tool in the proof of these results is the Beraha-Kahane-Weiss theorem on the limit sets of zeros for certain sequences of analytic functions, for which I give a new and simpler proof.Comment: LaTeX2e, 53 pages. Version 2 includes a new Appendix B. Version 3 adds a new Theorem 1.4 and a new Section 5, and makes several small improvements. To appear in Combinatorics, Probability & Computin

Conflict-free coloring of graphs

We study the conflict-free chromatic number chi_{CF} of graphs from extremal and probabilistic point of view. We resolve a question of Pach and Tardos about the maximum conflict-free chromatic number an n-vertex graph can have. Our construction is randomized. In relation to this we study the evolution of the conflict-free chromatic number of the Erd\H{o}s-R\'enyi random graph G(n,p) and give the asymptotics for p=omega(1/n). We also show that for p \geq 1/2 the conflict-free chromatic number differs from the domination number by at most 3.Comment: 12 page

A Strong Edge-Coloring of Graphs with Maximum Degree 4 Using 22 Colors

In 1985, Erd\H{o}s and Ne\'{s}etril conjectured that the strong edge-coloring number of a graph is bounded above by ${5/4}\Delta^2$ when $\Delta$ is even and ${1/4}(5\Delta^2-2\Delta+1)$ when $\Delta$ is odd. They gave a simple construction which requires this many colors. The conjecture has been verified for $\Delta\leq 3$. For $\Delta=4$, the conjectured bound is 20. Previously, the best known upper bound was 23 due to Horak. In this paper we give an algorithm that uses at most 22 colors.Comment: 9 pages, 4 figure