Search arXiv⌕ Search
Home/Archive/arXiv

arXiv · 1703.00959

The Hilton--Zhao Conjecture is True for Graphs with Maximum Degree 4

Daniel W. Cranston·Landon Rabern
View original source ↗Read source document ↗Export metadata ↓

Abstract

A simple graph $G$ is \emph{overfull} if $|E(G)|>Δ\lfloor|V(G)|/2\rfloor$. By the pigeonhole principle, every overfull graph $G$ has $χ'(G)>Δ$. The \emph{core} of a graph, denoted $G_Δ$, is the subgraph induced by its vertices of degree $Δ$. Vizing's Adjacency Lemma implies that if $χ'(G)>Δ$, then $G_Δ$ contains cycles. Hilton and Zhao conjectured that if $G_Δ$ has maximum degree 2 and $Δ\ge 4$, then $χ'(G)>Δ$ precisely when $G$ is overfull. We prove this conjecture for the case $Δ=4$.

Explore related subjects

math.CO

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Daniel W. Cranston, Landon Rabern. 2019-05-18. The Hilton--Zhao Conjecture is True for Graphs with Maximum Degree 4. https://arxiv.org/abs/1703.00959

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Orthogonal Pairs in Maps from the Sphere to the Circle

We prove that, for any $f:S^2\to S^1$ and any $\varepsilon>0$, there exist orthogonal vectors $x,y\in S^2$ such that the length of the shortest arc between $f(x)$ and $f(y)$ is at most $π/2 +\varepsilon$. This proves a conjecture of Ghebleh from 2007 that the circular chromatic number of the real orthogonality graph is equal to four.

math.CO↗
arXiv2026

Exact Area-Range Minima in the Quantitative Monsky Problem for Five and Seven Triangles

For a dissection $D$ of the unit square into $n$ nondegenerate triangles, let $R(D)=\max_i a_i-\min_i a_i, Δ(n)=\inf_D R(D).$ We prove that this infimum is attained for every $n\ge2$, and determine the exact minima for $n=5$ and $n=7$, allowing T-junctions. For five triangles, $Δ(5)=\frac{5\sqrt5-11}{8};$ equality holds precisely when three areas equal $(3-\sqrt5)/4$ and two equal $(3\sqrt5-5)/8$. For seven triangles, $Δ(7)=r_7$, where $r_7$ is the unique root in $(0,1/4900)$ of $864r^4+2160r^3-6060r^2+4972r-1.$ Every minimizer has four areas $(1+3r_7)/7$ and three areas $(1-4r_7)/7$, although its geometry need not be unique. The proofs combine finite combinatorial classification with exact symbolic and integer-interval certificates. For nine triangles, a tilted-strip construction gives the explicit algebraic upper bound $Δ(9)\le 0.0001273496861283553341\ldots,$ which is the exact minimum within that topology. Conversely, every dissection in the complete single-cap two-rail zig-zag family, with arbitrary continuous areas, has range greater than $1/3500$; hence a global minimizer must lie outside that family. The exact value of $Δ(9)$ remains open.

math.CO↗
arXiv2026

Chromatic symmetric functions for annular webs

We introduce a combinatorial definition of chromatic symmetric functions for annular webs. We prove their symmetry by constructing a web analogue of the Shareshian--Wachs involution and show that they coincide with the symmetric functions associated to annular webs via Turaev's isomorphism. We then derive explicit formulas for their hook Schur coefficients. We also introduce web LLT functions, whose hook Schur coefficients admit positive Laurent-polynomial formulas. These formulas yield a combinatorial expression for the coefficients of the HOMFLY--PT polynomial of an annular web.

math.CO↗