A Density Tur\'an Theorem
Lothar Narins, Tuan Tran

TL;DR
This paper extends a classical density Turán theorem to multipartite graphs, characterizing the minimum edge density needed to guarantee a subgraph, with specific structural conditions on the subgraph.
Contribution
It establishes a precise density threshold for multipartite graphs to contain a given subgraph, generalizing and extending previous results for cliques.
Findings
Determines the exact density threshold for multipartite graphs to contain a subgraph H.
Characterizes the structure of H for the threshold to hold, involving vertex coloring and matchings.
Recovers known results for cliques and extends to broader classes of graphs.
Abstract
Let be a graph which contains an edge whose deletion reduces its chromatic number. For such a graph a classical result of Simonovits from 1966 shows that every graph on vertices with more than edges contains a copy of . In this paper we derive a similar theorem for multipartite graphs. For a graph and an integer , let be the minimum real number such that every -partite graph whose edge density between any two parts is greater than contains a copy of . Our main contribution is to show that for sufficiently large if and only if admits a vertex-colouring with colours such that all colour classes but one are independent sets, and the exceptional class induces just a matching. When…
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Taxonomy
TopicsLimits and Structures in Graph Theory · Advanced Graph Theory Research
