Imbalance conjecture
OPG_edge_imbalance_multiset_graphic
Status solved high confidence
Schreib and Yavari prove that the positive imbalance multiset $M_G$ is graphic, establishing the imbalance conjecture in full.
Cited literature (3)
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Schreib and Yavari prove that the positive imbalance multiset $M_G$ is graphic, establishing the imbalance conjecture in full.
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Proves that certain unary and binary graph operations preserve graphicness of imbalance sequences, and discusses several conjectures related to graphicness, including connections between them.
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Establishes imbalance graphicness for unicyclic graphs, antiregular graphs, and three classes of block graphs, extending earlier results but leaving the general conjecture open.
Reviewer notes. These status corrections report results attributed to existing papers or to the final source version. Graph-Theory-LLM-Proofs located and checked the implication; it is not credited as the author of the result.
Discussion
Consider simple undirected graph $ G $ and let $ e=uv\in E(G) $ . The imbalance of the edge $ e $ defined as $ imb(e)=|deg(u)-deg(v)| $ . The multiset of all edge imbalances of $ G $ is denoted by $ M_{G} $ . Note, that conjecture is verified for all such graphs with $ \leq 9 $ vertices.
Bibliography
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[?]Graphs with graphic imbalance sequences Graphs with graphic imbalance sequences