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A complete solution to the biased Alon-Krivelevich-Spencer-Szabó criterion problem for the discrepancy game Scaffolds for Higher Tropical Grassmannians: Foundations Krahn-Szegő type inequalities for graphs $\boldsymbol{2}$-Neighbor Bootstrap Percolation on Odd Graphs Modular inequalities and Alexander polynomials of pencil type conic-line arrangements Counterexamples regarding elementary symmetric partitions Bounds on Multiple $b$-Burst Deletion-Correcting Codes The mex statistic on combinatorial structures Proper Additive Edge Colorings of Regular Graphs Off-diagonal Ramsey numbers Semi-inducibility of 4-vertex graphs A coarse Halin Grid Theorem with applications to quasi-transitive, locally finite graphs Bounds on the game isolation number and exact values for paths and cycles Minors of non-hamiltonian polyhedra and the Herschel family Triangle-free subsets of the $r$-distance graph of the hypercube Notes on sum-free sets in abelian groups When are Hopf algebras determined by integer sequences? A note on the irrationality of $ζ_2(5)$ Support-sensitive bounds for shortest zero-sum subsequences Weighted Random Dot Product Graphs More efficient sifting for grid norms, and applications to multiparty communication complexity Ordered Yao graphs: maximum degree, edge density, and clique numbers Refined enumeration of planar Eulerian orientations Real algebraic surfaces biholomorphically equivalent but not algebraically equivalent Bubble sort and Howe duality for staircase matrices Self-affine quadrangles On zero-sum Ramsey numbers modulo 3 Covering half-grids with lines and planes Borel Local Lemma: arbitrary random variables and limited exponential growth Dynamic capacity allocation of hybrid transportation units for cargo-hitching in urban public transportation systems An Efficient Genus Algorithm Based on Graph Rotations Neural Networks and (Virtual) Extended Formulations Some conjectures on $r$-graphs and equivalences Pointwise order of generalized Hofstadter functions G, H and beyond On the periodic decompositions of multidimensional configurations A q-analogue of graph independence polynomials with a group-theoretic interpretation Moment-angle manifolds corresponding to three-dimensional simplicial spheres, chordality and connected sums of products of spheres Strong Embeddings of 3-Connected Cubic Planar Graphs on Surfaces of non-negative Euler Characteristic Higher-rank trees arising from polyhedral graphs $L^p$ averages of the Fourier transform in finite fields A Bijection between Stacked Directed Polyominoes and Motzkin Paths with Alternative Catastrophes Open problems on relations of numerical semigroups Alcove walk models for parabolic Mirković-Vilonen intersections and branching to Levi subgroups A Mathematical Reconstruction of Endothelial Cell Networks Matchings in the hypercube with specified edges Geometric planted matchings beyond the Gaussian model Extensions and Deletions of matroid classes closed under flats String attractors and bi-infinite 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Specialization of Monomial Symmetric Polynomials and Group Determinants of Cyclic Groups Multidimensional Manhattan Preferences Balancing permuted copies of multigraphs and integer matrices Mixing times of one-sided $k$-transposition shuffles Embeddings of $k$-complexes in $2k$-manifolds and minimum rank of partial symmetric matrices The pre-Pieri rules Synchronization of strongly connected partial DFAs and prefix codes Large rainbow matchings in edge-colored graphs Surface cluster algebra expansion formulae via loop graphs Hit-and-Run for Sampling and Planning in Non-Convex Spaces Combinatorial Aspects of the Distribution of Rough Objects Lattice structure of Weyl groups via representation theory of preprojective algebras A Novel Paradigm for Calculating Ramsey Number via Artificial Bee Colony Algorithm A Cambrian framework for the oriented cycle Performance of the Survey Propagation-guided decimation algorithm for the random NAE-K-SAT problem A framework for large-scale distributed 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The Total Coloring Conjecture holds for planar graphs without three special subgraphs
Rongjin Su, Gang Fang, Enqiang Zhu · 2025-07-17 · via math.CO updates on arXiv.org

The Total Coloring Conjecture (TCC) for planar graphs with a maximum degree of six remains open. Previous studies suggest that TCC is valid for such graphs if they do not contain any subgraph isomorphic to a 4-fan. In this paper, we present an improved conclusion by establishing that TCC holds for planar graphs that are free of three particular substructures, namely the mushroom, the tent, and the cone. This advancement enhances previous findings by demonstrating that TCC is applicable to planar graphs with a maximum degree of six, which can accommodate sparse 4-fans, 5-fans, 5-wheels, and 6-wheels.

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