Temat: chromatic uniqueness

Skocz do pozycji: 1.

Tytuł:: Hypergraphs with Pendant Paths are not Chromatically Unique
Autorzy:: Tomescu, Ioan
Tematy:: sunflower hypergraph
chromatic polynomial
pendant path
chromatic uniqueness; Pokaż więcej
Wydawca:: Uniwersytet Zielonogórski. Wydział Matematyki, Informatyki i Ekonometrii
Powiązania:: https://bibliotekanauki.pl/articles/30147214.pdf Link otwiera się w nowym oknie
Opis:: In this note it is shown that every hypergraph containing a pendant path of length at least 2 is not chromatically unique. The same conclusion holds for h-uniform r-quasi linear 3-cycle if r ≥ 2.
Dostawca treści:: Biblioteka Nauki

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Tytuł:: Beta invariant and chromatic uniqueness of wheels
Autorzy:: Lee, Sooyeon
Wu, Haidong
Tematy:: chromatic uniqueness of graphs
beta invariant
characteristic polynomial
2-sum
3-sum
matroids; Pokaż więcej
Wydawca:: Uniwersytet Zielonogórski. Wydział Matematyki, Informatyki i Ekonometrii
Powiązania:: https://bibliotekanauki.pl/articles/59882201.pdf Link otwiera się w nowym oknie
Opis:: A graph $G$ is chromatically unique if its chromatic polynomial completely determines the graph. An $n$-spoked wheel, $W_n$, is shown to be chromatically unique when $n\ge 4$ is even [S.-J. Xu and N.-Z. Li, The chromaticity of wheels, Discrete Math. 51 (1984) 207–212]. When $n$ is odd, this problem is still open for $n\ge 15$ since 1984, although it was shown by different researchers that the answer is no for $n=5, 7$, yes for $n=3, 9, 11, 13$, and unknown for other odd $n$. We use the beta invariant of matroids to prove that if $M$ is a 3-connected matroid such that $|E(M)| = |E(W_n)|$ and $\beta(M) = \beta(M(W_n))$, where $\beta(M)$ is the beta invariant of $M$, then $M \cong M(W_n)$. As a consequence, if $G$ is a 3-connected graph such that the chromatic (or flow) polynomial of $G$ equals to the chromatic (or flow) polynomial of a wheel, then $G$ is isomorphic to the wheel. The examples for $n=3, 5$ show that the 3-connectedness condition may not be dropped. We also give a splitting formula for computing the beta invariants of general parallel connection of two matroids as well as the 3-sum of two binary matroids. This generalizes the corresponding result of Brylawski [A combinatorial model for series-parallel networks, Trans. Amer. Math. Soc. 154 (1971) 1–22].
Dostawca treści:: Biblioteka Nauki

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