On the existence of zero-sum perfect matchings of complete graphs

Kittipassorn, Teeradej; Sinsap, Panon

doi:10.26493/1855-3974.2573.90d

Cited by 3 publications

(2 citation statements)

References 11 publications

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“…The first, third and fourth authors studied the balanceability of several graph classes [8]. In [2,9,13], a closely related problem is studied for spanning balanced subgraphs. In [1], colorings with arbitrary many colors are studied and the corresponding 3-color balancing number of paths is determined upon a constant factor.…”

Section: Introductionmentioning

confidence: 99%

The Balancing Number and Generalized Balancing Number of Some Graph Classes

Dailly¹,

Eslava²,

Hansberg³

et al. 2023

Electron. J. Combin.

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Given a graph $G$, a 2-coloring of the edges of $K_n$ is said to contain a balanced copy of $G$ if we can find a copy of $G$ such that half of its edges is in each color class. If there exists an integer $k$ such that, for $n$ sufficiently large, every 2-coloring of $K_n$ with more than $k$ edges in each color contains a balanced copy of $G$, then we say that $G$ is balanceable. The smallest integer $k$ such that this holds is called the balancing number of $G$.In this paper, we define a more general variant of the balancing number, the generalized balancing number, by considering 2-coverings of the edge set of $K_n$, where every edge $e$ has an associated list $L(e)$ which is a nonempty subset of the color set $\{r,b\}$. In this case, edges $e$ with $L(e) = \{r,b\}$ act as jokers in the sense that their color can be chosen $r$ or $b$ as needed. In contrast to the balancing number, every graph has a generalized balancing number. Moreover, if the balancing number exists, then it coincides with the generalized balancing number.We give the exact value of the generalized balancing number for all cycles except for cycles of length $4k$ for which we give tight bounds. In addition, we give general bounds for the generalized balancing number of non-balanceable graphs based on the extremal number of its subgraphs, and study the generalized balancing number of $K_5$, which turns out to be surprisingly large.

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Section: Introductionmentioning

confidence: 99%

The Balancing Number and Generalized Balancing Number of Some Graph Classes

Dailly¹,

Eslava²,

Hansberg³

et al. 2023

Electron. J. Combin.

View full text Add to dashboard Cite

show abstract

“…In the case that c is balanced, that is, d = 1 2 , the existence of copies G π for which |c(G π )| is small, or equivalently, m + (G π ) is close to m(G) 2 has been studied under the term zero sum problems or zero sum Ramsey theory; see [4,5,6,7,9,14,15,16,17] as well as the references therein pointing out connections to further research directions. The observations (1) and ( 2) are based on common arguments from this area, and together they imply the existence of some permutation π from S n with…”

Section: Introductionmentioning

confidence: 99%

Unbalanced Spanning Subgraphs in Edge Labeled Complete Graphs

Bessy¹,

Pardey²,

Picasarri-Arrieta³

et al. 2023

Electron. J. Combin.

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Let $K$ be a complete graph of order $n$. For $d\in (0,1)$, let $c$ be a $\pm 1$-edge labeling of $K$ such that there are $d{n\choose 2}$ edges with label $+1$, and let $G$ be a spanning subgraph of $K$ of maximum degree at most $\Delta$ and with $m(G)$ edges. We prove the existence of an isomorphic copy $G'$ of $G$ in $K$ such that the number of edges with label $+1$ in $G'$ is at least $\left(d+\frac{\min\left\{ 2-d-2\sqrt{1-d},\sqrt{d}-d\right\}}{2\Delta+1}-O\left(\frac{1}{n}\right)\right)m(G)$, that is, this number visibly exceeds its expected value $d\cdot m(G)$ when considering a uniformly random copy of $G$ in $K$. For $d=\frac{1}{2}$, and $\Delta\leq 2$, we present more detailed results.

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Efficiently finding low-sum copies of spanning forests in zero-sum complete graphs via conditional expectation

Pardey

Rautenbach

2023

Discrete Applied Mathematics

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On the existence of zero-sum perfect matchings of complete graphs

Cited by 3 publications

References 11 publications

The Balancing Number and Generalized Balancing Number of Some Graph Classes

The Balancing Number and Generalized Balancing Number of Some Graph Classes

Unbalanced Spanning Subgraphs in Edge Labeled Complete Graphs

Efficiently finding low-sum copies of spanning forests in zero-sum complete graphs via conditional expectation

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