Subgraph discrepancies in the complete graph
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866912874880827392 |
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| author | Christoph, Micha Gishboliner, Lior Krivelevich, Michael |
| author_facet | Christoph, Micha Gishboliner, Lior Krivelevich, Michael |
| contents | Given a 2-edge-coloring $f : E(K_n) \rightarrow \{\pm 1\}$, the discrepancy of a subgraph $F \subseteq K_n$ is defined as $\left| \sum_{e \in E(F)} f(e) \right|$. Erdős, Füredi, Loebl and Sós showed that if $F$ is an $n$-vertex tree with maximum degree at most $(1-\varepsilon)n$, then every 2-coloring of $K_n$ has a copy of $F$ with discrepancy $Ω(\varepsilon)n$. We extend this result by showing that the same conclusion holds for every $n$-vertex graph with maximum degree at most $(1-\varepsilon)n$ and no isolated vertices. We also show that for every $d$-regular $n$-vertex graph $F$ with $d \leq (1-\varepsilon)n$, every 2-coloring of $K_n$ has a copy of $F$ with discrepancy $Ω(\sqrt{\varepsilon d}) \cdot n$. The dependence on $d$ and $n$ is best possible.
Finally, we consider specific graphs $F$, namely $K_r$-factors and 2-factors. For each such graph $F$, we determine the optimal constant $λ$ such that every 2-coloring of $K_n$ has a copy of $F$ with discrepancy at least $(λ+ o(1))n$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2602_04069 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Subgraph discrepancies in the complete graph Christoph, Micha Gishboliner, Lior Krivelevich, Michael Combinatorics Given a 2-edge-coloring $f : E(K_n) \rightarrow \{\pm 1\}$, the discrepancy of a subgraph $F \subseteq K_n$ is defined as $\left| \sum_{e \in E(F)} f(e) \right|$. Erdős, Füredi, Loebl and Sós showed that if $F$ is an $n$-vertex tree with maximum degree at most $(1-\varepsilon)n$, then every 2-coloring of $K_n$ has a copy of $F$ with discrepancy $Ω(\varepsilon)n$. We extend this result by showing that the same conclusion holds for every $n$-vertex graph with maximum degree at most $(1-\varepsilon)n$ and no isolated vertices. We also show that for every $d$-regular $n$-vertex graph $F$ with $d \leq (1-\varepsilon)n$, every 2-coloring of $K_n$ has a copy of $F$ with discrepancy $Ω(\sqrt{\varepsilon d}) \cdot n$. The dependence on $d$ and $n$ is best possible. Finally, we consider specific graphs $F$, namely $K_r$-factors and 2-factors. For each such graph $F$, we determine the optimal constant $λ$ such that every 2-coloring of $K_n$ has a copy of $F$ with discrepancy at least $(λ+ o(1))n$. |
| title | Subgraph discrepancies in the complete graph |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2602.04069 |