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We study how many comparability subgraphs are needed to partition the edge set of a perfect graph. We show that many classes of perfect graphs can be partitioned into (at most) two comparability subgraphs and this holds for almost all perfect graphs. On the other hand, we prove that for interval graphs an arbitrarily large number of comparability subgraphs might be necessary.

We call a proper edge coloring of a graph $G$ a B-coloring if every 4-cycle of $G$ is colored with four different colors. Let $q_B(G)$ denote the smallest number of colors needed for a B-coloring of $G$. Motivated by earlier papers on B-colorings, here we consider $q_B(G)$ for planar and outerplanar graphs in terms of the maximum degree $\Delta = \Delta(G)$. We prove that $q_B(G)\le 2\Delta+8$ for planar graphs, $q_B(G)\le 2\Delta$ for bipartite planar graphs and $q_B(G)\le \Delta+1$ for outerplanar graphs with $\Delta \ge 4$. We conjecture that, for $\Delta$ sufficiently large, $q_B(G)\le 2\Delta(G)$ for planar $G$ and $q_B(G)\le \Delta(G)$ for outerplanar $G$., Comment: 15 pages, 4 figures, to appear in Journal of Graph Theory

Assume that $R_1,R_2,\dots,R_t$ are disjoint parallel lines in the plane. A $t$-interval (or $t$-track interval) is a set that can be written as the union of $t$ closed intervals, each on a different line. It is known that pairwise intersecting $2$-intervals can be pierced by two points, one from each line. However, it is not true that every set of pairwise intersecting $3$-intervals can be pierced by three points, one from each line. For $k\ge 3$, Kaiser and Rabinovich asked whether $k$-wise intersecting $t$-intervals can be pierced by $t$ points, one from each line. Our main result provides an asymptotic answer: in any set $S_1,\dots,S_n$ of $k$-wise intersecting $t$-intervals, at least $\frac{k-1}{k+1}n$ can be pierced by $t$ points, one from each line. We prove this in a more general form, replacing intervals by subtrees of a tree. This leads to questions and results on covering vertices of edge-colored complete graphs by vertices of monochromatic cliques having distinct colors, where the colorings are chordal, or more generally induced $C_4$-free graphs. For instance, we show that if the edges of a complete graph $K_n$ are colored with red or blue so that both color classes are induced $C_4$-free, then at least ${4n\over 5}$ vertices can be covered by a red and a blue clique, and this is best possible. We conclude by pointing to new Ramsey-type problems emerging from these restricted colorings., Comment: 15 pages, 3 figures

Two independent edges in ordered graphs can be nested, crossing or separated. These relations define six types of subgraphs, depending on which relations are forbidden. We refine a remark by Erd\H{o}s and Rado that every 2-coloring of the edges of a complete graph contains a monochromatic spanning tree. We show that forbidding one relation we always have a monochromatic (non-nested, non-crossing, non-separated) spanning tree in a 2-edge-colored ordered complete graph. On the other hand, if two relations are forbidden, then it is possible that we have monochromatic (nested, separated, crossing) subtrees of size only half the number of vertices. The existence of a monochromatic non-nested spanning tree in 2-colorings of ordered complete graphs verifies a more general conjecture for twisted drawings. Our second subject is to refine the Ramsey number of matchings for ordered complete graphs. Cockayne and Lorimer proved that for given positive integers t, n, m=(t-1)(n-1)+2n is the smallest integer with the following property: every t-coloring of the edges of a complete graph Km contains a monochromatic matching with n edges. We conjecture a strengthening: t-colored ordered complete graphs on m vertices contain monochromatic non-nested and also non-separated matchings with n edges. We prove this conjecture for some special cases. (i) Every t-colored ordered complete graph on t+3 vertices contains a monochromatic non-nested matching of size two. (ii) Every 2-colored ordered complete graph on 3n-1 vertices contains a monochromatic non-separated matching of size n. For nested, separated, and crossing matchings the situation is different. The smallest m ensuring a monochromatic matching of size n in every t-coloring is 2(t(n-1))+1) in the first two cases and one less in the third case.