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Families of Linearly $\chi$-bounded Graphs without Chair or its Induced Sub-graphs

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Medha Dhurandhar

Combinatorics · Submitted 27 Dec 2023

abstract compiled · 2026-09-23 · no language-model review

The authors' main claim Here we prove that if a graph $G$ is free of 1. {Chair; P$_4$+K$_1$} or 2. {Chair; HVN}, then $\chi(G)$ is linearly bounded by maximum clique size of G.

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arXiv ID2312.16399 (version 1)
Pith IDEU6U4H6E, Pith's permanent record of this paper and its review
CitationMedha Dhurandhar. "Families of Linearly $\chi$-bounded Graphs without Chair or its Induced Sub-graphs." arXiv:2312.16399 (2023). https://pith.science/paper/EU6U4H6E
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Compiled by classifier-1.13.0 via TypeSafe API, question bank abstract-p1-20260923, 2026-09-23: 1 request, 3,488 input tokens.

What the abstract itself states, as probabilities: claims first 0.03 · states limitation 0.08 · human participants 0.01 · controlled comparison 0.01 · randomized assignment 0.01 · states quantitative result 0.90 · compares with prior results 0.06 · mentions code or data availability 0.01.

The abstract, sentence by sentence

arXiv listing abstract v1. Results and the main claim are underlined; .

A hereditary class H of graphs is $\chi$-bounded if there is a $\chi$-binding function f such that for every $G$ in $H$, $\chi(G)$ less than or equal to $f(\omega(G))$. Here we prove that if a graph $G$ is free of 1. {Chair; P$_4$+K$_1$} or 2. {Chair; HVN}, then $\chi(G)$ is linearly bounded by maximum clique size of G. We further prove that if $G$ is free of 3. {P$_4$+K$_1$; P$_3$ $\cup$ K$_1$} or 4. {P4+K1; K$_2$ $\cup$ 2K$_1$} or 5. {HVN; P$_3$ $\cup$ K$_1$} or 6. {HVN; K$_2$ $\cup$ 2K$_1$} or 7. {K$_{5-e}$; P$_3$ $\cup$ K$_1$} or 8. {K$_{5-e}$; K$_2$ $\cup$ 2K$_1$}, then there is a tight linear $\chi$-bound for $G$.

background contribution method result interpretation main claim

What the classifier reports for each sentence. The number is the probability it gave the label.
#sentencerolethe authors' own workstates the main conclusionstrength
s0 A hereditary class H of graphs is $\chi$-bounded if there is a $\chi$-binding function f such that for every $G$ in $H$, $\chi(G)$ less than or equal to $f(\omega(G))$. background 1.00 0.06 0.02 asserted 0.98
s1 Here we prove that if a graph $G$ is free of 1. {Chair; P$_4$+K$_1$} or 2. {Chair; HVN}, then $\chi(G)$ is linearly bounded by maximum clique size of G. result 0.91 0.96 0.64 asserted 1.00
s2 We further prove that if $G$ is free of 3. {P$_4$+K$_1$; P$_3$ $\cup$ K$_1$} or 4. {P4+K1; K$_2$ $\cup$ 2K$_1$} or 5. {HVN; P$_3$ $\cup$ K$_1$} or 6. {HVN; K$_2$ $\cup$ 2K$_1$} or 7. {K$_{5-e}$; P$_3$ $\cup$ K$_1$} or 8. {K$_{5-e}$; K$_2$ $\cup$ 2K$_1$}, then there is a tight linear $\chi$-bound for $G$. result 0.88 0.95 0.27 asserted 1.00

What kind of paper the classifier reports this is: theoretical result 1.00 · evidence from proof or derivation 1.00.

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Every quoted sentence on this record is the authors'. Every label is from Pith's closed vocabulary, with the probability shown beside it. Every number in the measurements table is the authors', checked by code. Record as JSON

The paper
Original abstract

A hereditary class H of graphs is $\chi$-bounded if there is a $\chi$-binding function f such that for every $G$ in $H$, $\chi(G)$ less than or equal to $f(\omega(G))$. Here we prove that if a graph $G$ is free of 1. {Chair; P$_4$+K$_1$} or 2. {Chair; HVN}, then $\chi(G)$ is linearly bounded by maximum clique size of G. We further prove that if $G$ is free of 3. {P$_4$+K$_1$; P$_3$ $\cup$ K$_1$} or 4. {P4+K1; K$_2$ $\cup$ 2K$_1$} or 5. {HVN; P$_3$ $\cup$ K$_1$} or 6. {HVN; K$_2$ $\cup$ 2K$_1$} or 7. {K$_{5-e}$; P$_3$ $\cup$ K$_1$} or 8. {K$_{5-e}$; K$_2$ $\cup$ 2K$_1$}, then there is a tight linear $\chi$-bound for $G$.

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