The chromatic discrepancy of graphs

For a proper vertex coloring cc of a graph GG, let φc(G)φc(G) denote the maximum, over all induced subgraphs HH of GG, the difference between the chromatic number χ(H)χ(H) and the number of colors used by cc to color HH. We define the chromatic discrepancy of a graph GG, denoted by φ(G)φ(G), to be the minimum φc(G)φc(G), over all proper colorings cc of GG. If HH is restricted to only connected induced subgraphs, we denote the corresponding parameter by View the MathML sourceφˆ(G). These parameters are aimed at studying graph colorings that use as few colors as possible in a graph and all its induced subgraphs. We study the parameters φ(G)φ(G) and View the MathML sourceφˆ(G) and obtain bounds on them. We obtain general bounds, as well as bounds for certain special classes of graphs including random graphs. We provide structural characterizations of graphs with φ(G)=0φ(G)=0 and graphs with View the MathML sourceφˆ(G)=0. We also show that computing these parameters is NP-hard

Structure and colour in triangle-free graphs

Motivated by a recent conjecture of the first author, we prove that every properly coloured triangle-free graph of chromatic number $\chi$ contains a rainbow independent set of size $\lceil\frac12\chi\rceil$. This is sharp up to a factor $2$. This result and its short proof have implications for the related notion of chromatic discrepancy. Drawing inspiration from both structural and extremal graph theory, we conjecture that every triangle-free graph of chromatic number $\chi$ contains an induced cycle of length $\Omega(\chi\log\chi)$ as $\chi\to\infty$. Even if one only demands an induced path of length $\Omega(\chi\log\chi)$, the conclusion would be sharp up to a constant multiple. We prove it for regular girth $5$ graphs and for girth $21$ graphs. As a common strengthening of the induced paths form of this conjecture and of Johansson's theorem (1996), we posit the existence of some $c >0$ such that for every forest $H$ on $D$ vertices, every triangle-free and induced $H$-free graph has chromatic number at most $c D/\log D$. We prove this assertion with `triangle-free' replaced by `regular girth $5$'.Comment: 12 pages; in v2 one section was removed due to a subtle erro

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

Blow up and Blur constructions in Algebraic Logic

The idea in the title is to blow up a finite structure, replacing each 'colour or atom' by infinitely many, using blurs to represent the resulting term algebra, but the blurs are not enough to blur the structure of the finite structure in the complex algebra. Then, the latter cannot be representable due to a {finite- infinite} contradiction. This structure can be a finite clique in a graph or a finite relation algebra or a finite cylindric algebra. This theme gives examples of weakly representable atom structures that are not strongly representable. Many constructions existing in the literature are placed in a rigorous way in such a framework, properly defined. This is the essence too of construction of Monk like-algebras, one constructs graphs with finite colouring (finitely many blurs), converging to one with infinitely many, so that the original algebra is also blurred at the complex algebra level, and the term algebra is completey representable, yielding a representation of its completion the complex algebra. A reverse of this process exists in the literature, it builds algebras with infinite blurs converging to one with finite blurs. This idea due to Hirsch and Hodkinson, uses probabilistic methods of Erdos to construct a sequence of graphs with infinite chromatic number one that is 2 colourable. This construction, which works for both relation and cylindric algebras, further shows that the class of strongly representable atom structures is not elementary.Comment: arXiv admin note: text overlap with arXiv:1304.114