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An edge‐ordered graph is a graph with a linear ordering of its edges. Two edge‐ordered graphs areequivalentif there is an isomorphism between them preserving the edge‐ordering. Theedge‐ordered Ramsey number redge(H; q) of an edge‐ordered graphHis the smallestNsuch that there exists an edge‐ordered graphGonNvertices such that, for everyq‐coloring of the edges ofG, there is a monochromatic subgraph ofGequivalent toH. Recently, Balko and Vizer proved thatredge(H; q) exists, but their proof gave enormous upper bounds on these numbers. We give a new proof with a much better bound, showing there exists a constantcsuch thatfor every edge‐ordered graphHonnvertices. We also prove a polynomial bound for the edge‐ordered Ramsey number of graphs of bounded degeneracy. Finally, we prove a strengthening for graphs where every edge has a label and the labels do not necessarily have an ordering.
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Degree of h-polynomials of edge ideals
Abstract In this paper, we investigate the degree ofh-polynomials of edge ideals of finite simple graphs. In particular, we provide combinatorial formulas for the degree of theh-polynomial for various fundamental classes of graphs such as paths, cycles, and bipartite graphs. To the best of our knowledge, this study represents the first investigation into the combinatorial interpretation of this algebraic invariant. Additionally, we characterize all connected graphs in which the sum of the Castelnuovo–Mumford regularity and the degree of theh-polynomial of an edge ideal achieve its maximum value, equal to the number of vertices in the graph.
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- Award ID(s):
- 2418805
- PAR ID:
- 10630902
- Publisher / Repository:
- Springer Science + Business Media
- Date Published:
- Journal Name:
- Journal of Algebraic Combinatorics
- Volume:
- 62
- Issue:
- 1
- ISSN:
- 0925-9899
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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