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Variable degeneracy on toroidal graphs

Rui Li, Tao Wang

TL;DR

The work develops a unified framework for strictly $f$-degenerate transversals as a generalization of DP-coloring and $L$-forested-coloring, and applies it to toroidal and planar graphs via forbidden configurations. It introduces valued covers, building covers, and monoblocks, and leverages discharging to establish structural lemmas that guarantee the existence of strictly $f$-degenerate transversals under a universal $f$-sum condition of at least $4$. Consequently, it proves that toroidal graphs without certain subgraphs (NOA or $4$-cycles) are DP-$4$-colorable and have list vertex arboricity at most $2$, with analogous planar-graph results. These findings extend DP-coloring techniques to broader surface-embedded graphs and improve prior bounds on DP-$4$-coloring and list vertex arboricity.

Abstract

DP-coloring was introduced by Dvořák and Postle as a generalization of list coloring and signed coloring. A new coloring, strictly $f$-degenerate transversal, is a further generalization of DP-coloring and $L$-forested-coloring. In this paper, we present some structural results on planar and toroidal graphs with forbidden configurations, and establish some sufficient conditions for the existence of strictly $f$-degenerate transversal based on these structural results. Consequently, (i) every toroidal graph without subgraphs isomorphic to the configurations in Fig.2 is DP-$4$-colorable, and has list vertex arboricity at most $2$, (ii) every toroidal graph without $4$-cycles is DP-$4$-colorable, and has list vertex arboricity at most $2$, (iii) every planar graph without subgraphs isomorphic to the configurations in Fig.3 is DP-$4$-colorable, and has list vertex arboricity at most $2$. These results improve upon previous results on DP-$4$-coloring [Discrete Math. 341~(7) (2018) 1983--1986; Bull. Malays. Math. Sci. Soc. 43~(3) (2020) 2271--2285] and (list) vertex arboricity [Discrete Math. 333 (2014) 101--105; Int. J. Math. Stat. 16~(1) (2015) 97--105; Iranian Math. Soc. 42~(5) (2016) 1293--1303].

Variable degeneracy on toroidal graphs

TL;DR

The work develops a unified framework for strictly -degenerate transversals as a generalization of DP-coloring and -forested-coloring, and applies it to toroidal and planar graphs via forbidden configurations. It introduces valued covers, building covers, and monoblocks, and leverages discharging to establish structural lemmas that guarantee the existence of strictly -degenerate transversals under a universal -sum condition of at least . Consequently, it proves that toroidal graphs without certain subgraphs (NOA or -cycles) are DP--colorable and have list vertex arboricity at most , with analogous planar-graph results. These findings extend DP-coloring techniques to broader surface-embedded graphs and improve prior bounds on DP--coloring and list vertex arboricity.

Abstract

DP-coloring was introduced by Dvořák and Postle as a generalization of list coloring and signed coloring. A new coloring, strictly -degenerate transversal, is a further generalization of DP-coloring and -forested-coloring. In this paper, we present some structural results on planar and toroidal graphs with forbidden configurations, and establish some sufficient conditions for the existence of strictly -degenerate transversal based on these structural results. Consequently, (i) every toroidal graph without subgraphs isomorphic to the configurations in Fig.2 is DP--colorable, and has list vertex arboricity at most , (ii) every toroidal graph without -cycles is DP--colorable, and has list vertex arboricity at most , (iii) every planar graph without subgraphs isomorphic to the configurations in Fig.3 is DP--colorable, and has list vertex arboricity at most . These results improve upon previous results on DP--coloring [Discrete Math. 341~(7) (2018) 1983--1986; Bull. Malays. Math. Sci. Soc. 43~(3) (2020) 2271--2285] and (list) vertex arboricity [Discrete Math. 333 (2014) 101--105; Int. J. Math. Stat. 16~(1) (2015) 97--105; Iranian Math. Soc. 42~(5) (2016) 1293--1303].

Paper Structure

This paper contains 4 sections, 23 theorems, 1 equation, 4 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Results on toroidal graphs
  4. Results on planar graphs

Key Result

Theorem 1

A connected toroidal graph that does not include subgraphs isomorphic to the configurations as depicted in NOA, has minimum degree at most $3$, unless it is a $2$-connected $4$-regular graph with Euler characteristic $\epsilon(G) = 0$.

Figures (4)

  • Figure 1: The $5$-cap-subgraph.
  • Figure 2: Forbidden configurations in \ref{['MLONE', 'MRONEPLANAR', 'MRONE']}.
  • Figure 3: Forbidden configurations in \ref{['MLTHREE', 'MRTHREE']}.
  • Figure 4: Forbidden configurations in \ref{['F35-Subgraph']}.

Theorems & Definitions (29)

  • Theorem 1
  • Theorem 2
  • Theorem 3
  • Theorem 4: Choi and Zhang MR3233411
  • Theorem 5
  • Theorem 6
  • Theorem 7
  • Proposition 1: Lu et al. MR4357325
  • Corollary 1
  • Definition 1
  • ...and 19 more