In the mid-1990s, Stanley and Stembridge conjectured that the chromatic symmetric functions of claw-free co-comparability (also called incomparability) graphs were e-positive. The quest for the proof of this conjecture has led to an examination of other, related graph classes. In 2013 Guay-Paquet proved that if unit interval graphs are e-positive, that implies claw-free incomparability graphs are as well. Inspired by this approach, we consider a related case and prove that unit interval graphs whose complement is also a unit interval graph are e-positive. We introduce the concept of strongly e-positive to denote a graph whose induced subgraphs are all e-positive, and conjecture that a graph is strongly e-positive if and only if it is (claw, net)-free. * Formerly Angèle M. Hamel. Precup [7] have proved e-positivity for several subclasses of unit interval graphs. The time is ripe for further investigations of subclasses and superclasses of unit interval graphs.Graphs and their complements are natural pairs to study. The (claw, co-claw)-free graphs hold particular interest. Two of the authors investigated them in [8], concluding they were not all e-positive. Here we revisit this result, showing that the particular graph called the net is the only exception. This result follows by careful consideration of the graph structure, and subsequent decomposition into constituent graphs. From this analysis, along with a number of powerful graph theory results, we derive a series of results, culminating in a theorem that states that if a graph G and its complement are both unit interval graphs, then G is e-positive.The graph class universe we are working in is captured by Figure 1. The class of claw-free co-comparability graphs targeted by Stanley and Stembridge wholly contains the subclass of unit interval graphs. If we look at the larger picture we see that the superclass of claw-free, AT-free graphs (see definition of AT-free in Section 2) consists of co-triangle-free graphs (known to be e-positive [15], restated in Theorem 2.3) and claw-free co-comparability graphs. Thus proving the Stanley and Stembridge conjecture would prove all claw-free, AT-free graphs were e-positive.Even farther beyond this is the class of (claw, net)-free graphs. The net (see Figure 2) is significant as this is the example originally given by Stanley [16] of a claw-free, non-e-positive, graph to show claw-free alone is not a property sufficient to guarantee epositivity. We focus particularly on (claw, net)-free graphs (note that for n = 4 there is one non-e-positive graph (namely, the claw, K 1,3 ), for n = 5 there are 4 non-e-positive connected graphs (namely K 1,4 , dart, cricket = K 1,4 + e, co-{K 3 ∪ 2K 1 }), for n = 6, there are 44 non-e-positive connected graphs, and for n = 7 there are 374 non-e-positive connected graphs). To our knowledge, this paper is the first exploration of the (claw, net)free e-positivity question. We conjecture these graphs are e-positive. We have verified our conjecture for graphs up to 9 vertices. We also introduc...
In this paper we show that every ( P 6, diamond)‐free graph G satisfies χ ( G ) ≤ ω ( G ) + 3, where χ ( G ) and ω ( G ) are the chromatic number and clique number of G, respectively. Our bound is attained by the complement of the famous 27‐vertex Schläfli graph. Our result unifies previously known results on the existence of linear χ‐binding functions for several graph classes. Our proof is based on a reduction via the Strong Perfect Graph Theorem to imperfect ( P 6, diamond)‐free graphs, a careful analysis of the structure of those graphs, and a computer search that relies on a well‐known characterization of 3‐colourable ( P 6 , K 3 )‐free graphs.
As usual, $P_{n}$ ($n\geq 1$) denotes the path on $n$ vertices. The gem is the graph consisting of a $P_{4}$ together with an additional vertex adjacent to each vertex of the $P_{4}$. A graph is called ($P_{5}$, gem)-free if it has no induced subgraph isomorphic to a $P_{5}$ or to a gem. For a graph $G$, $\unicode[STIX]{x1D712}(G)$ denotes its chromatic number and $\unicode[STIX]{x1D714}(G)$ denotes the maximum size of a clique in $G$. We show that $\unicode[STIX]{x1D712}(G)\leq \lfloor \frac{3}{2}\unicode[STIX]{x1D714}(G)\rfloor$ for every ($P_{5}$, gem)-free graph $G$.
The reconfiguration graph for the k-colourings of a graph G, denoted R k (G), is the graph whose vertices are the k-colourings of G and two colourings are joined by an edge if they differ in colour on exactly one vertex. For any k-colourable P 4 -free graph G, Bonamy and Bousquet proved that R k+1 (G) is connected. In this short note, we complete the classification of the connectedness of R k+1 (G) for a k-colourable graph G excluding a fixed path, by constructing a 7chromatic 2K 2 -free (and hence P 5 -free) graph admitting a frozen 8colouring. This settles a question of the second author.
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