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Chromatic Symmetric Functions and Hfree Graphs Hamel, Angele
Description
A key area of investigation in chromatic symmetric functions concerns the epositivity, and/or Schur positivity, of a particular class of chromatic symmetric functions whose underlying graphs are clawfree. The interest in these graphs originates in a paper of Stanley and Stembridge (On immanants of JacobiTrudi matrices and permutations with restricted position, JCTA 62 (1993), 261279) that considers positivity of coefficients of immanants. A natural consideration in their study is a conjecture that states that if a poset is (3+1)free, then its incomparability graph, which is necessarily clawfree, is epositive. As a related result, Gasharov (Incomparability graphs of (3+1)free posets are spositive, Disc. Math. 157 (1996), 193197) proved that they are Schur positive. Stanley (Graph colorings and related symmetric functions: ideas and applications, Disc. Math. 193 (1998), 267286.) has further conjectured that *any* clawfree graph is Schur positive. In parallel to this, in graph theory, much effort has been spent in characterizing the chromatic characteristics of graphs that are Hfree, where H is some set of induced subgraphs. A key question in this domain is, can the chromatic number of a Hfree graph be determined in polynomial time? This answer is known for large classes of graphs, but, interestingly, the classification for all combinations of subgraphs with 4 vertices is almost, but not quite, complete, Lozin and Malyshev (Vertex coloring of graphs with few obstructions, Disc. Appl. Math. 216 (2017), 273280). The claw is a graph on 4 vertices. Do the other graphs on 4 vertices have similar positivity results? In graph theory when it is too hard to say something about a single subgraph, the question of multiple subgraphs is often considered (e.g. if we cannot derive results about clawfree graphs, perhaps we can derive results about (claw, 4cycle)free graphs), hence the positivity question can similarly be made easier by asking about a set of subgraphs rather than a single one. Just as the question about clawfree graphs was interesting because of the connection to immanant conjectures, so too would questions about other 4 vertex graphs be interesting because of the parallel with studies in graph theory.
Item Metadata
Title 
Chromatic Symmetric Functions and Hfree Graphs

Creator  
Publisher 
Banff International Research Station for Mathematical Innovation and Discovery

Date Issued 
20170515T11:01

Description 
A key area of investigation in chromatic symmetric functions concerns the epositivity, and/or Schur positivity, of a particular class of chromatic symmetric functions whose underlying graphs are clawfree. The interest in these graphs originates in a paper of Stanley and Stembridge (On immanants of JacobiTrudi matrices and permutations with restricted position, JCTA 62 (1993), 261279) that considers positivity of coefficients of immanants. A natural consideration in their study is a conjecture that states that if a poset is (3+1)free, then its incomparability graph, which is necessarily clawfree, is epositive. As a related result, Gasharov (Incomparability graphs of (3+1)free posets are spositive, Disc. Math. 157 (1996), 193197) proved that they are Schur positive. Stanley (Graph colorings and related symmetric functions: ideas and applications, Disc. Math. 193 (1998), 267286.) has further conjectured that *any* clawfree graph is Schur positive.
In parallel to this, in graph theory, much effort has been spent in characterizing the chromatic characteristics of graphs that are Hfree, where H is some set of induced subgraphs. A key question in this domain is, can the chromatic number of a Hfree graph be determined in polynomial time? This answer is known for large classes of graphs, but, interestingly, the classification for all combinations of subgraphs with 4 vertices is almost, but not quite, complete, Lozin and Malyshev (Vertex coloring of graphs with few obstructions, Disc. Appl. Math. 216 (2017), 273280).
The claw is a graph on 4 vertices. Do the other graphs on 4 vertices have similar positivity results? In graph theory when it is too hard to say something about a single subgraph, the question of multiple subgraphs is often considered (e.g. if we cannot derive results about clawfree graphs, perhaps we can derive results about (claw, 4cycle)free graphs), hence the positivity question can similarly be made easier by asking about a set of subgraphs rather than a single one. Just as the question about clawfree graphs was interesting because of the connection to immanant conjectures, so too would questions about other 4 vertex graphs be interesting because of the parallel with studies in graph theory.

Extent 
22 minutes

Subject  
Type  
File Format 
video/mp4

Language 
eng

Notes 
Author affiliation: Wilfrid Laurier University

Series  
Date Available 
20171111

Provider 
Vancouver : University of British Columbia Library

Rights 
AttributionNonCommercialNoDerivatives 4.0 International

DOI 
10.14288/1.0357928

URI  
Affiliation  
Peer Review Status 
Unreviewed

Scholarly Level 
Faculty

Rights URI  
Aggregated Source Repository 
DSpace

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Item Citations and Data
Rights
AttributionNonCommercialNoDerivatives 4.0 International