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# Cheolwon Heo (허철원), Representations of even-cycle matroids

## Tuesday, August 31, 2021 @ 4:30 PM - 5:30 PM KST

Room B232, IBS (기초과학연구원)

### Speaker

Cheolwon Heo (허철원)
Applied Algebra and Optimization Research Center (AORC), Sungkyunkwan University

A signed graph is a pair $(G,\Sigma)$ where $G$ is a graph and $\Sigma$ is a subset of edges of $G$. A cycle $C$ of $G$ is a subset of edges of $G$ such that every vertex of the subgraph of $G$ induced by $C$ has an even degree. We say that $C$ is even in $(G,\Sigma)$ if $|C \cap \Sigma|$ is even; otherwise, $C$ is odd. A matroid $M$ is an even-cycle matroid if there exists a signed graph $(G,\Sigma)$ such that circuits of $M$ precisely corresponds to inclusion-wise minimal non-empty even cycles of $(G,\Sigma)$. For even-cycle matroids, two fundamental questions arise:
(1) what is the relationship between two signed graphs representing the same even-cycle matroids?
(2) how many signed graphs can an even-cycle matroid have?
For (a), we characterize two signed graphs $(G_1,\Sigma_1)$ and $(G_2,\Sigma_2)$ where $G_1$ and $G_2$ are $4$-connected that represent the same even-cycle matroids.
For (b), we introduce pinch-graphic matroids, which can generate exponentially many representations even when the matroid is $3$-connected. An even-cycle matroid is a pinch-graphic matroid if there exists a signed graph with a pair of vertices such that every odd cycle intersects with at least one of them. We prove that there exists a constant $c$ such that if a matroid is even-cycle matroid that is not pinch-graphic, then the number of representations is bounded by $c$. This is joint work with Bertrand Guenin and Irene Pivotto.

## Details

Date:
Tuesday, August 31, 2021
Time:
4:30 PM - 5:30 PM KST
Event Category:
Event Tags:
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Room B232
IBS (기초과학연구원)

## Organizer

Sang-il Oum (엄상일)
View Organizer Website 기초과학연구원 수리및계산과학연구단 이산수학그룹
대전 유성구 엑스포로 55 (우) 34126
IBS Discrete Mathematics Group (DIMAG)
Institute for Basic Science (IBS)
55 Expo-ro Yuseong-gu Daejeon 34126 South Korea
E-mail: dimag@ibs.re.kr, Fax: +82-42-878-9209