NONUNIFORM DEGREES AND RAINBOW VERSIONS OF THE CACCETTA-HÄGGKVIST CONJECTURE

Ron Aharoni; Eli Berger; Maria Chudnovsky; He Guo; Shira Zerbib

doi:10.1137/22M1529658

NONUNIFORM DEGREES AND RAINBOW VERSIONS OF THE CACCETTA-HÄGGKVIST CONJECTURE

Ron Aharoni, Eli Berger, Maria Chudnovsky, He Guo, Shira Zerbib

Department of Mathematics

Research output: Contribution to journal › Article › peer-review

Abstract

The Caccetta-Häggkvist conjecture (denoted CHC) states that the directed girth (the smallest length of a directed cycle) dgirth(D) of a directed graph D on n vertices is at most (equation presented)⌈ⁿ_δ+(D)⌉, where δ⁺(D) is the minimum outdegree of D. We consider a version involving all outdegrees, not merely the minimum one, and prove that if D does not contain a sink, then dgirth(D) ≤ 2 (equation presented)-_{v∊V(D) deg+}¹_(v)+1. In the spirit of a generalization of the CHC to rainbow cycles in [1], this suggests the conjecture that given nonempty sets F₁,..., F_n of edges of K_n, there exists a rainbow cycle of length at most 2 (equation presented)-_1≤i≤n |F_i¹ | ₊₁. We prove a bit stronger result when 1 ≤ | F_i| ≤ 2, thereby strengthening a result of DeVos et al. [J. Graph Theory, 96 (2021), pp. 192-202]. We prove a logarithmic bound on the rainbow girth in the case that the sets Fi are triangles.

Original language	English
Pages (from-to)	1704-1714
Number of pages	11
Journal	SIAM Journal on Discrete Mathematics
Volume	37
Issue number	3
DOIs	https://doi.org/10.1137/22M1529658
State	Published - 2023

Bibliographical note

Publisher Copyright:
© 2023 He Guo.

Keywords

directed girth
generalized Caccetta-Häggkvist conjecture
rainbow girth

ASJC Scopus subject areas

General Mathematics

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10.1137/22M1529658

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title = "NONUNIFORM DEGREES AND RAINBOW VERSIONS OF THE CACCETTA-H{\"A}GGKVIST CONJECTURE",

abstract = "The Caccetta-H{\"a}ggkvist conjecture (denoted CHC) states that the directed girth (the smallest length of a directed cycle) dgirth(D) of a directed graph D on n vertices is at most (equation presented)⌈nδ+(D)⌉, where δ+(D) is the minimum outdegree of D. We consider a version involving all outdegrees, not merely the minimum one, and prove that if D does not contain a sink, then dgirth(D) ≤ 2 (equation presented)-v∊V(D) deg+1(v)+1. In the spirit of a generalization of the CHC to rainbow cycles in [1], this suggests the conjecture that given nonempty sets F1,..., Fn of edges of Kn, there exists a rainbow cycle of length at most 2 (equation presented)-1≤i≤n |Fi1 | +1. We prove a bit stronger result when 1 ≤ | Fi| ≤ 2, thereby strengthening a result of DeVos et al. [J. Graph Theory, 96 (2021), pp. 192-202]. We prove a logarithmic bound on the rainbow girth in the case that the sets Fi are triangles.",

keywords = "directed girth, generalized Caccetta-H{\"a}ggkvist conjecture, rainbow girth",

author = "Ron Aharoni and Eli Berger and Maria Chudnovsky and He Guo and Shira Zerbib",

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year = "2023",

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AU - Guo, He

AU - Zerbib, Shira

PY - 2023

Y1 - 2023

N2 - The Caccetta-Häggkvist conjecture (denoted CHC) states that the directed girth (the smallest length of a directed cycle) dgirth(D) of a directed graph D on n vertices is at most (equation presented)⌈nδ+(D)⌉, where δ+(D) is the minimum outdegree of D. We consider a version involving all outdegrees, not merely the minimum one, and prove that if D does not contain a sink, then dgirth(D) ≤ 2 (equation presented)-v∊V(D) deg+1(v)+1. In the spirit of a generalization of the CHC to rainbow cycles in [1], this suggests the conjecture that given nonempty sets F1,..., Fn of edges of Kn, there exists a rainbow cycle of length at most 2 (equation presented)-1≤i≤n |Fi1 | +1. We prove a bit stronger result when 1 ≤ | Fi| ≤ 2, thereby strengthening a result of DeVos et al. [J. Graph Theory, 96 (2021), pp. 192-202]. We prove a logarithmic bound on the rainbow girth in the case that the sets Fi are triangles.

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NONUNIFORM DEGREES AND RAINBOW VERSIONS OF THE CACCETTA-HÄGGKVIST CONJECTURE

Abstract

Bibliographical note

Keywords

ASJC Scopus subject areas

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