On the square coloring of comparability graphs

We find sufficient conditions for the square of a comparability graph [Formula: see text] of a poset [Formula: see text] to be [Formula: see text]-colorable when [Formula: see text] lacks [Formula: see text] for some [Formula: see text]. Furthermore, we show that the problem of coloring the square of the comparability graph of a poset of height at least four can be reduced to the case of height three, where the height of a poset is the size of a maximum chain.

Representing Graphs via Pattern Avoiding Words

The notion of a word-representable graph has been studied in a series of papers in the literature. A graph $G=(V,E)$ is word-representable if there exists a word $w$ over the alphabet $V$ such that letters $x$ and $y$ alternate in $w$ if and only if $xy$ is an edge in $E$. If $V =\{1, \ldots, n\}$, this is equivalent to saying that $G$ is word-representable if for all $x,y \in \{1, \ldots, n\}$, $xy \in E$ if and only if the subword $w_{\{x,y\}}$ of $w$ consisting of all occurrences of $x$ or $y$ in $w$ has no consecutive occurrence of the pattern 11.In this paper, we introduce the study of $u$-representable graphs for any word $u \in \{1,2\}^*$. A graph $G$ is $u$-representable if and only if there is a vertex-labeled version of $G$, $G=(\{1, \ldots, n\}, E)$, and a word $w \in \{1, \ldots, n\}^*$ such that for all $x,y \in \{1, \ldots, n\}$, $xy \in E$ if and only if $w_{\{x,y\}}$ has no consecutive occurrence of the pattern $u$. Thus, word-representable graphs are just $11$-representable graphs. We show that for any $k \geq 3$, every finite graph $G$ is $1^k$-representable. This contrasts with the fact that not all graphs are 11-representable graphs.The main focus of the paper is the study of $12$-representable graphs. In particular, we classify the $12$-representable trees. We show that any $12$-representable graph is a comparability graph and the class of $12$-representable graphs include the classes of co-interval graphs and permutation graphs. We also state a number of facts on $12$-representation of induced subgraphs of a grid graph.

Some Discrete Optimization Problems With Hamming and H-Comparability Graphs

Any H-comparability graph contains a Hamming graph as spanningsubgraph. An acyclic orientation of an H-comparability graph contains an acyclic orientation of the spanning Hamming graph, called sequence graph in the classical open-shop scheduling problem. We formulate different discrete optimization problems on the Hamming graphs and on H-comparability graphs and consider their complexity and relationship. Moreover, we explore the structures of these graphs in the class of irreducible sequences for the open shop problem in this paper.

Edge ideals of clique clutters of comparability graphs and the normality of monomial ideals

The normality of a monomial ideal is expressed in terms of lattice points of blocking polyhedra and the integer decomposition property. For edge ideals of clutters this property characterizes normality. Let $G$ be the comparability graph of a finite poset. If $\mathrm{cl}(G)$ is the clutter of maximal cliques of $G$, we prove that $\mathrm{cl}(G)$ satisfies the max-flow min-cut property and that its edge ideal is normally torsion free. Then we prove that edge ideals of complete admissible uniform clutters are normally torsion free.