Searcharxiv⌕ Search
Home/Archive/arXiv

arXiv · 0707.4580

Note on edge-colored graphs and digraphs without properly colored cycles

Gregory Gutin
View original source ↗Read source document ↗Export metadata ↓

Abstract

We study the following two functions: d(n,c) and $\vec{d}(n,c)$; d(n,c) ($\vec{d}(n,c)$) is the minimum number k such that every c-edge-colored undirected (directed) graph of order n and minimum monochromatic degree (out-degree) at least k has a properly colored cycle. Abouelaoualim et al. (2007) stated a conjecture which implies that d(n,c)=1. Using a recursive construction of c-edge-colored graphs with minimum monochromatic degree p and without properly colored cycles, we show that $d(n,c)\ge {1 \over c}(\log_cn -\log_c\log_cn)$ and, thus, the conjecture does not hold. In particular, this inequality significantly improves a lower bound on $\vec{d}(n,2)$ obtained by Gutin, Sudakov and Yeo in 1998.

Explore related subjects

cs.DM

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Gregory Gutin. 2007-07-31. Note on edge-colored graphs and digraphs without properly colored cycles. https://arxiv.org/abs/0707.4580

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

KEEP EXPLORING

Related papers

arXiv2026

Ternary Binomial and Trinomial Bent Functions in the Completed Maiorana-McFarland Class

Two classes of ternary (vectorial) bent functions of degree four with two and three terms in the univariate representation that belong to the completed Maiorana-McFarland original class are found. Binomials are mappings $\F_{3^{4k}}\mapsto\fthreek$ given by $\Tr^{4k}_k\big(a_1 x^{2(3^k+1)}+a_2 x^{(3^k+1)^2}\big)$, where $a_1$ is a nonsquare in $\F_{3^{4k}}$ and $a_2$ is defined explicitly by $a_1$. Particular subclass of such binomial vectorial bent function is represented by the exceptional polynomials over $\fthreek$. Bent trinomials are mappings $\F_{3^{2k}}\mapsto\fthree$ given by $\Tr_{2k}\big(a_1 x^{2(3^k+2)} + a_2 x^{3^k+5} + a_3 x^2\big)$ with $k>1$ not divisible by four and coefficients explicitly defined depending on the parity of $k$. The proof is based on new criterion that allows checking bentness by analyzing the first-order derivatives of $f$ in the direction of a chosen $n/2$-dimensional subspace ($n$ is the degree of the domain of $f$ over $\fp$).

cs.DM↗
arXiv2026

Minimizing the makespan in job shop scheduling under conflict graph constraints

We study the job shop scheduling problem with a conflict graph, in which adjacent jobs cannot be processed simultaneously on different machines, with the objective of minimizing the makespan. We establish a polynomial equivalence with a resource-constrained job shop problem having unit-capacity resources and binary resource requirements The problem is strongly NP-hard even on two machines with unit-time operations. For this setting, we derive an optimal-makespan formula and a linear-time algorithm when the conflict graph is a cluster graph, given the clique partition. We also develop a polynomial-time $4/3$-approximation algorithm for arbitrary conflict graphs on two machines when each job has exactly two unit-time operations. For the general problem, we propose four mixed-integer linear formulations, lower bounds, and a genetic algorithm with active, non-delay, and hybrid decoding procedures. Computational experiments on instances derived from the Lawrence and Taillard benchmarks and on randomly generated generalized job shop instances evaluate the proposed methods and show that the genetic algorithm generally improves on constructive heuristics.

cs.DM↗
arXiv2026

A Linear-Time Algorithm for the Unbiased Claw Game on Graphs with Maximum Degree 4

We study the unbiased Maker-Breaker $K_{1,3}$-game, the unbiased claw game, played on the edge set of a general graph $G$. In this game, Maker and Breaker alternatingly claim edges of $G$, and Maker wins if his claimed edges contain a claw once no more edges are available. Graphs with maximum degree at least 5 are Maker's win. A graph $G$ is claw-connected if the hypergraph modeling the unbiased claw game played on $G$ is connected. We characterize Maker's win in the unbiased claw game on every claw-connected graph whose maximum degree is different from 3. Our main structural result is a complete characterization of Maker's win for claw-connected graphs with exactly one vertex of degree 4. The analysis is based on the fact that vertices of degree 4 act as forcing structures. Our characterization results in a linear-time algorithm for the unbiased claw game on claw-connected graphs with maximum degree different from 3. We also obtain a linear-time decomposition algorithm reducing the game on general graphs to a collection of pairwise edge-disjoint claw-connected subgraphs of maximum degree exactly 3, thereby isolating the only remaining case. This decomposition yields an $\mathcal{O}(|V(G)|^3)$-time algorithm for the unbiased claw game on general graphs, improving on the $\mathcal{O}(|V(G)|^7)$-time algorithm implied by the work of Galliot, Gravier and Sivignon (arXiv 2026). For a lower bound on the decision complexity, we prove that any agent deciding the game in the adjacency-array model makes at least $Ω(|V(G)|)$ queries in the worst case, indicating the optimality of our linear-time algorithm.

cs.DM↗