Algebraic Shifting of Finite Graphs

Abstract

In the present article, for bipartite graphs and chordal graphs, their exterior algebraic shifted graph and their symmetric algebraic shifted graph are studied. First, we will determine the symmetric algebraic shifted graph of complete bipartite graphs. It turns out that for a ≥ 3 and b ≥ 3, the exterior algebraic shifted graph of the complete bipartite graph K _a,b of size a, b is different from the symmetric algebraic shifted graph of K _a,b . Second, we will show that the exterior algebraic shifted graph of any chordal graph G coincides with the symmetric algebraic shifted graph of G. In addition, it will be shown that the exterior algebraic shifted graph of any chordal graph G is equal to some combinatorial shifted graph of G.

Key Words:

• Algebraic shifting
• Bipartite graphs
• Chordal graphs
• Generic initial ideals

AMS Subject Classification:

• 13F55
• 05C75
• 05C85
• 13D02

ACKNOWLEDGMENT

The author is supported by JSPS Research Fellowships for Young Scientists.

Notes

Communicated by I. Swanson.