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For a graph $H$, a graph $G$ is $H$-induced-saturated if $G$ does not contain an induced copy of $H$, but either removing an edge from $G$ or adding a non-edge to $G$ creates an induced copy of $H$. Depending on the graph $H$, an $H$-induced-saturated 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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For a fixed graph $F$ and an integer $t$, the dfn{rainbow saturation number} of $F$, denoted by $sat_t(n,mathfrak{R}(F))$, is defined as the minimum number of edges in a $t$-edge-colored graph on $n$ vertices which does not contain a dfn{rainbow copy} of $F$, i.e., a copy of $F$ all of whose edges receive a different color, but the addition of any missing edge in any color from $[t]$ creates such a rainbow copy. Barrus, Ferrara, Vardenbussche and Wenger prove that $sat_t(n,mathfrak{R}(P_ell))ge n-1$ for $ellge 4$ and $sat_t(n,mathfrak{R}(P_ell))le lceil frac{n}{ell-1} rceil cdot binom{ell-1}{2}$ for $tge binom{ell-1}{2}$, where $P_ell$ is a path with $ell$ edges. In this short note, we improve the upper bounds and show that $sat_t(n,mathfrak{R}(P_ell))le lceil frac{n}{ell} rceil cdot left({{ell-2}choose {2}}+4right)$ for $ellge 5$ and $tge 2ell-5$.
We present a modification of the DFS graph search algorithm, suited for finding long induced paths. We use it to give simple proofs of the following results. We show that the induced size-Ramsey number of paths satisfies $hat{R}_{mathrm{ind}}(P_n)leq 5cdot 10^7n$, thus giving an explicit constant in the linear bound, improving the previous bound with a large constant from a regularity lemma argument by Haxell, Kohayakawa and {L}uczak. We also provide a bound for the $k$-color version, showing that $hat{R}_{mathrm{ind}}^k(P_n)=O(k^3log^4k)n$. Finally, we present a new short proof of the fact that the binomial random graph in the supercritical regime, $G(n,frac{1+varepsilon}{n})$, contains typically an induced path of length $Theta(varepsilon^2) n$.
Given a graph $G=(V,E)$ whose vertices have been properly coloured, we say that a path in $G$ is colourful if no two vertices in the path have the same colour. It is a corollary of the Gallai-Roy-Vitaver Theorem that every properly coloured graph contains a colourful path on $chi(G)$ vertices. We explore a conjecture that states that every properly coloured triangle-free graph $G$ contains an induced colourful path on $chi(G)$ vertices and prove its correctness when the girth of $G$ is at least $chi(G)$. Recent work on this conjecture by Gyarfas and Sarkozy, and Scott and Seymour has shown the existence of a function $f$ such that if $chi(G)geq f(k)$, then an induced colourful path on $k$ vertices is guaranteed to exist in any properly coloured triangle-free graph $G$.
We show that for $dge d_0(epsilon)$, with high probability, the random graph $G(n,d/n)$ contains an induced path of length $(3/2-epsilon)frac{n}{d}log d$. This improves a result obtained independently by Luczak and Suen in the early 90s, and answers a question of Fernandez de la Vega. Along the way, we generalize a recent result of Cooley, Draganic, Kang and Sudakov who studied the analogous problem for induced matchings.
It is an open problem whether the 3-coloring problem can be solved in polynomial time in the class of graphs that do not contain an induced path on $t$ vertices, for fixed $t$. We propose an algorithm that, given a 3-colorable graph without an induced path on $t$ vertices, computes a coloring with $max{5,2lceil{frac{t-1}{2}}rceil-2}$ many colors. If the input graph is triangle-free, we only need $max{4,lceil{frac{t-1}{2}}rceil+1}$ many colors. The running time of our algorithm is $O((3^{t-2}+t^2)m+n)$ if the input graph has $n$ vertices and $m$ edges.
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