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تسجيل مستخدم جديدF-Saturation Games
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We study F-saturation games, first introduced by Furedi, Reimer and Seress in 1991, and named as such by West. The main question is to determine the length of the game whilst avoiding various classes of graph, playing on a large complete graph. We show lower bounds on the length of path-avoiding games, and more precise results for short paths. We show sharp results for the tree avoiding game and the star avoiding game.
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We study analogues of $mathcal{F}$-saturation games, first introduced by Furedi, Reimer and Seress in 1991, and named as such by West. We examine analogous games on directed graphs, and show tight results on the walk-avoiding game. We also examine an
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Given a family of graphs $mathcal{F}$, we define the $mathcal{F}$-saturation game as follows. Two players alternate adding edges to an initially empty graph on $n$ vertices, with the only constraint being that neither player can add an edge that crea
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d graph does not necessarily exist. In fact, Martin and Smith (2012) showed that $P_4$-induced-saturated graphs do not exist, where $P_k$ denotes a path on $k$ vertices. Axenovich and Csik{o}s (2019) asked the existence of $P_k$-induced-saturated graphs for $k ge 5$; it is easy to construct such graphs when $kin{2, 3}$. Recently, R{a}ty constructed a graph that is $P_6$-induced-saturated. In this paper, we show that there exists a $P_{k}$-induced-saturated graph for infinitely many values of $k$. To be precise, we find a $P_{3n}$-induced-saturated graph for every positive integer $n$. As a consequence, for each positive integer $n$, we construct infinitely many $P_{3n}$-induced-saturated graphs. We also show that the Kneser graph $K(n,2)$ is $P_6$-induced-saturated for every $nge 5$.
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hcal{C}$. Similarly to classical saturation functions, define $mathrm{sat}_t(n, mathcal{C})$ to be the minimum number of edges in a $(mathcal{C},t)$ saturated graph. Let $mathcal{C}_r(H)$ be the family consisting of every edge-colored copy of $H$ which uses exactly $r$ colors. In this paper we consider a variety of colored saturation problems. We determine the order of magnitude for $mathrm{sat}_t(n, mathcal{C}_r(K_k))$ for all $r$, showing a sharp change in behavior when $rgeq binom{k-1}{2}+2$. A particular case of this theorem proves a conjecture of Barrus, Ferrara, Vandenbussche, and Wenger. We determine $mathrm{sat}_t(n, mathcal{C}_2(K_3))$ exactly and determine the extremal graphs. Additionally, we document some interesting irregularities in the colored saturation function.
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