Search arXiv⌕ Search
Home/Archive/arXiv

arXiv · 1710.01472

A new upper bound on the game chromatic index of graphs

Ralph Keusch
View original source ↗Read source document ↗Export metadata ↓

Abstract

We study the two-player game where Maker and Breaker alternately color the edges of a given graph $G$ with $k$ colors such that adjacent edges never get the same color. Maker's goal is to play such that at the end of the game, all edges are colored. Vice-versa, Breaker wins as soon as there is an uncolored edge where every color is blocked. The game chromatic index $χ'_g(G)$ denotes the smallest $k$ for which Maker has a winning strategy. The trivial bounds $Δ(G) \le χ_g'(G) \le 2Δ(G)-1$ hold for every graph $G$, where $Δ(G)$ is the maximum degree of $G$. In 2008, Beveridge, Bohman, Frieze, and Pikhurko proved that for every $δ>0$ there exists a constant $c>0$ such that $χ'_g(G) \le (2-c)Δ(G)$ holds for any graph with $Δ(G) \ge (\frac{1}{2}+δ)v(G)$, and conjectured that the same holds for every graph $G$. In this paper, we show that $χ'_g(G) \le (2-c)Δ(G)$ is true for all graphs $G$ with $Δ(G) \ge C \log v(G)$. In addition, we consider a biased version of the game where Breaker is allowed to color $b$ edges per turn and give bounds on the number of colors needed for Maker to win this biased game.

Explore related subjects

math.CO

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Ralph Keusch. 2018-02-13. A new upper bound on the game chromatic index of graphs. https://arxiv.org/abs/1710.01472

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↗