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The Erd\H{o}s-Simonovits stability theorem states that for all \epsilon >0 there exists \alpha >0 such that if G is a K_{r+1}-free graph on n vertices with e(G) > ex(n,K_{r+1}) - \alpha n^2, then one can remove \epsilon n^2 edges from G to obtain an r-partite graph. F\"uredi gave a short proof that one can choose \alpha=\epsilon. We give a bound for the relationship of \alpha and \varepsilon which is asymptotically sharp as \epsilon \to 0., Comment: 12 pages, 1 figure

For a real constant $\alpha$, let $\pi_3^\alpha(G)$ be the minimum of twice the number of $K_2$'s plus $\alpha$ times the number of $K_3$'s over all edge decompositions of $G$ into copies of $K_2$ and $K_3$, where $K_r$ denotes the complete graph on $r$ vertices. Let $\pi_3^\alpha(n)$ be the maximum of $\pi_3^\alpha(G)$ over all graphs $G$ with $n$ vertices. The extremal function $\pi_3^3(n)$ was first studied by Gy\H{o}ri and Tuza [Decompositions of graphs into complete subgraphs of given order, Studia Sci. Math. Hungar. 22 (1987), 315--320]. In a recent progress on this problem, Kr\'al', Lidick\'y, Martins and Pehova [Decomposing graphs into edges and triangles, Combin. Prob. Comput. 28 (2019) 465--472] proved via flag algebras that $\pi_3^3(n)\le (1/2+o(1))n^2$. We extend their result by determining the exact value of $\pi_3^\alpha(n)$ and the set of extremal graphs for all $\alpha$ and sufficiently large $n$. In particular, we show for $\alpha=3$ that $K_n$ and the complete bipartite graph $K_{\lfloor n/2\rfloor,\lceil n/2\rceil}$ are the only possible extremal examples for large $n$., Comment: 20 pages, 3 figures

For all $n\ge 9$, we show that the only triangle-free graphs on $n$ vertices maximizing the number $5$-cycles are balanced blow-ups of a 5-cycle. This completely resolves a conjecture by Erd\H{o}s, and extends results by Grzesik and Hatami, Hladk\'y, Kr\'{a}l', Norin and Razborov, where they independently showed this same result for large $n$ and for all $n$ divisible by $5$., Comment: 6 pages

A packing $k$-coloring for some integer $k$ of a graph $G=(V,E)$ is a mapping $\varphi:V\to\{1,\ldots,k\}$ such that any two vertices $u, v$ of color $\varphi(u)=\varphi(v)$ are in distance at least $\varphi(u)+1$. This concept is motivated by frequency assignment problems. The \emph{packing chromatic number} of $G$ is the smallest $k$ such that there exists a packing $k$-coloring of $G$. Fiala and Golovach showed that determining the packing chromatic number for chordal graphs is \NP-complete for diameter exactly 5. While the problem is easy to solve for diameter 2, we show \NP-completeness for any diameter at least 3. Our reduction also shows that the packing chromatic number is hard to approximate within $n^{{1/2}-\varepsilon}$ for any $\varepsilon > 0$. In addition, we design an \FPT algorithm for interval graphs of bounded diameter. This leads us to exploring the problem of finding a partial coloring that maximizes the number of colored vertices., Comment: 9 pages, 2 figures

A long standing open problem in extremal graph theory is to describe all graphs that maximize the number of induced copies of a path on four vertices. The character of the problem changes in the setting of oriented graphs, and becomes more tractable. Here we resolve this problem in the setting of oriented graphs without transitive triangles., Comment: 17 pages, 4 figures

The Hall ratio of a graph $G$ is the maximum value of $v(H) / \alpha(H)$ taken over all non-null subgraphs $H$ of $G$. For any graph, the Hall ratio is a lower-bound on its fractional chromatic number. In this note, we present various constructions of graphs whose fractional chromatic number grows much faster than their Hall ratio. This refutes a conjecture of Harris.

Let $\mathcal{C}$ be a family of edge-colored graphs. A $t$-edge colored graph $G$ is $(\mathcal{C}, t)$-saturated if $G$ does not contain any graph in $\mathcal{C}$ but the addition of any edge in any color in $[t]$ creates a copy of some graph in $\mathcal{C}$. 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 $r\geq \binom{k-1}{2}+2$. 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.

Finding exact Ramsey numbers is a problem typically restricted to relatively small graphs. The flag algebra method was developed to find asymptotic results for very large graphs, so it seems that the method is not suitable for finding small Ramsey numbers. But this intuition is wrong, and we will develop a technique to do just that in this paper. We find new upper bounds for many small graph and hypergraph Ramsey numbers. As a result, we prove the exact values $R(K_4^-,K_4^-,K_4^-)=28$, $R(K_8,C_5)= 29$, $R(K_9,C_6)= 41$, $R(Q_3,Q_3)=13$, $R(K_{3,5},K_{1,6})=17$, $R(C_3, C_5, C_5)= 17$, and $R(K_4^-,K_5^-;3)= 12$. We hope that this technique will be adapted to address other questions for smaller graphs with the flag algebra method., Comment: 17 pages, 2 figures

Turan's Theorem states that every graph of a certain edge density contains a complete graph $K^k$ and describes the unique extremal graphs. We give a similar Theorem for l-partite graphs. For large l, we find the minimal edge density $d^k_l$, such that every $\ell$-partite graph whose parts have pairwise edge density greater than $d^k_l$ contains a $K^k$. It turns out that $d^k_l=(k-2)/(k-1)$ for large enough l. We also describe the structure of the extremal graphs. For the case of triangles we show that $d^3_{13}=1/2$, disproving a conjecture by Bondy, Shen, Thomasse and Thomassen., Comment: 9 pages

Let $m:=|E(G)|$ sufficiently large and $s:=(m-1)/3$. We show that unless the maximum degree $\Delta > 2s$, there is a weighting $w:E\cup V\to \{0,1,...,s\}$ so that $w(uv)+w(u)+w(v)\ne w(u'v')+w(u')+w(v')$ whenever $uv\ne u'v'$ (such a weighting is called {\em total edge irregular}). This validates a conjecture by Ivanco and Jendrol' for large graphs, extending a result by Brandt, Miskuf and Rautenbach., Comment: 14 pages

Motivated by investigations of rainbow matchings in edge colored graphs, we introduce the notion of color-line graphs that generalizes the classical concept of line graphs in a natural way. Let $H$ be a (properly) edge-colored graph. The (proper) color-line graph $C\!L(H)$ of $H$ has edges of $H$ as vertices, and two edges of $H$ are adjacent in $C\!L(H)$ if they are incident in $H$ or have the same color. We give Krausz-type characterizations for (proper) color-line graphs, and point out that, for any fixed $k\ge 2$, recognizing if a graph is the color-line graph of some graph $H$ in which the edges are colored with at most $k$ colors is NP-complete. In contrast, we show that, for any fixed $k$, recognizing color-line graphs of properly edge colored graphs $H$ with at most $k$ colors is polynomially. Moreover, we give a good characterization for proper $2$-color line graphs that yields a linear time recognition algorithm in this case., To appear in Discrete Appl. Math

Erdos and Sos proposed a problem of determining the maximum number F(n) of rainbow triangles in 3-edge-colored complete graphs on n vertices. They conjectured that F(n) = F(a)+ F(b)+F(c)+F(d)+abc+abd+acd+bcd, where a+b+c+d = n and a, b, c, d are as equal as possible. We prove that the conjectured recurrence holds for sufficiently large n. We also prove the conjecture for n = 4k for all k. These results imply that lim F(n) n^3/6 = 0.4, and determine the unique limit object. In the proof we use flag algebras combined with stability arguments., 27 pages

Let $G$ be a fixed graph and let ${\mathcal F}$ be a family of graphs. A subgraph $J$ of $G$ is ${\mathcal F}$-saturated if no member of ${\mathcal F}$ is a subgraph of $J$, but for any edge $e$ in $E(G)-E(J)$, some element of ${\mathcal F}$ is a subgraph of $J+e$. We let $\text{ex}({\mathcal F},G)$ and $\text{sat}({\mathcal F},G)$ denote the maximum and minimum size of an ${\mathcal F}$-saturated subgraph of $G$, respectively. If no element of ${\mathcal F}$ is a subgraph of $G$, then $\text{sat}({\mathcal F},G) = \text{ex}({\mathcal F}, G) = |E(G)|$. In this paper, for $k\ge 3$ and $n\ge 100$ we determine $\text{sat}(K_3,K_k^n)$, where $K_k^n$ is the complete balanced $k$-partite graph with partite sets of size $n$. We also give several families of constructions of $K_t$-saturated subgraphs of $K_k^n$ for $t\ge 4$. Our results and constructions provide an informative contrast to recent results on the edge-density version of $\text{ex}(K_t,K_k^n)$ from [A. Bondy, J. Shen, S. Thomass\'e, and C. Thomassen, Density conditions for triangles in multipartite graphs, Combinatorica 26 (2006), 121--131] and [F. Pfender, Complete subgraphs in multipartite graphs, Combinatorica 32 (2012), no. 4, 483--495]., Comment: 16 pages, 4 figures

The crossing number cr(G) of a graph G is the minimum number of crossings in a nondegenerate planar drawing of G. The rectilinear crossing number cr'(G) of G is the minimum number of crossings in a rectilinear nondegenerate planar drawing (with edges as straight line segments) of G. Zarankiewicz proved in 1952 that cr'(K_{n_1,n_2})\le Z(n_1,n_2):= n_1/2*(n_1-1)/2*n_2/2*(n_2-1)/2. We define an analogous bound A(n_1,n_2,n_3) for the complete tripartite graph K_{n_1,n_2,n_3}, and prove that cr'(K_{n_1,n_2,n_3})\le A({n_1,n_2,n_3}). We also show that for n large enough, 0.973 A(n,n,n) \le cr'(K_{n,n,n}) and 0.666 A(n,n,n)\le cr(K_{n,n,n}), with the tighter rectilinear lower bound established through the use of flag algebras. A complete multipartite graph is balanced if the partite sets all have the same cardinality. We study asymptotic behavior of the crossing number of the balanced complete r-partite graph. Richter and Thomassen proved in 1997 that the limit as n\to\infty of cr(K_{n,n}) over the maximum number of crossings in a drawing of K_{n,n} exists and is at most 1/4. We define z(r)=3(r^2-r)/8(r^2+r-3) and show that for a fixed r and the balanced complete r-partite graph, z(r) is an upper bound to the limit superior of the crossing number divided by the maximum number of crossings in a drawing.

A rainbow matching in an edge-colored graph is a matching whose edges have distinct colors. We address the complexity issue of the following problem, \mrbm: Given an edge-colored graph $G$, how large is the largest rainbow matching in $G$? We present several sharp contrasts in the complexity of this problem. We show, among others, that * can be approximated by a polynomial algorithm with approximation ratio $2/3-\eps$. * is APX-complete, even when restricted to properly edge-colored linear forests without a $5$-vertex path, and is solvable in %time $O(m^{3/2})$ on edge-colored $m$-edge polynomial time for edge-colored forests without a $4$-vertex path. * is APX-complete, even when restricted to properly edge-colored trees without an $8$-vertex path, and is solvable in %time $O(n^{7/2})$ on edge-colored $n$-vertex polynomial time for edge-colored trees without a $7$-vertex path. * is APX-complete, even when restricted to properly edge-colored paths. These results provide a dichotomy theorem for the complexity of the problem on forests and trees in terms of forbidding paths. The latter is somewhat surprising, since, to the best of our knowledge, no (unweighted) graph problem prior to our result is known to be NP-hard for simple paths. We also address the parameterized complexity of the problem., To appear in Theoretical Computer Science

In 1959, Goodman determined the minimum number of monochromatic triangles in a complete graph whose edge set is two-coloured. Goodman also raised the question of proving analogous results for complete graphs whose edge sets are coloured with more than two colours. In this paper, we determine the minimum number of monochromatic triangles and the colourings which achieve this minimum in a sufficiently large three-coloured complete graph., Comment: Some data needed to verify the proof can be found at http://www.math.cmu.edu/users/jcumming/ckpsty/

For a given multigraph $H$, a graph $G$ is $H$-linked if $|G|\ge |H|$ and for every injective map $\tau: V(H)\to V(G)$, we can find internally disjoint paths in $G$, such that every edge from $uv$ in $H$ corresponds to a $\tau(u)-\tau(v)$ path. To guarantee that a $G$ is $H$-linked, you need a minimum degree larger than $\frac{|G|}{2}$. This situation changes, if you know that $G$ has a certain connectivity $k$. Depending on $k$, even a minimum degree independent of $|G|$ may suffice. Let $\delta(k,H,N)$ be the minimum number such that every $k$-connected graph $G$ with $|G|=N$ and $\delta(G)\ge \delta(k,H,N)$ is $H$-linked. We study bounds for this quantity. In particular, we find bounds for all multigraphs $H$ with at most three edges and some with four edges, which are optimal up to small additive or multiplicative constants. As a consequence, we also establish a few pure connectivity bounds for graph linkages.

For a graph $G$, let $t(G)$ denote the maximum number of vertices in an induced subgraph of $G$ that is a tree. Further, for a vertex $v\in V(G)$, let $t^v(G)$ denote the maximum number of vertices in an induced subgraph of $G$ that is a tree, with the extra condition that the tree must contain $v$. The minimum of $t(G)$ ($t^v(G)$, respectively) over all connected triangle-free graphs $G$ (and vertices $v\in V(G)$) on $n$ vertices is denoted by $t_3(n)$ ($t_3^v(n)$). Clearly, $t^v(G)\le t(G)$ for all $v\in V(G)$. In this note, we solve the extremal problem of maximizing $|G|$ for given $t^v(G)$, given that $G$ is connected and triangle-free. We show that $|G|\le 1+\frac{(t_v(G)-1)t_v(G)}{2}$ and determine the unique extremal graphs. Thus, we get as corollary that $t_3(n)\ge t_3^v(n)=\lceil {1/2}(1+\sqrt{8n-7})\rceil$, improving a recent result by Fox, Loh and Sudakov., Comment: 2 pages, minor edits for better readability

We present an extension of the Delsarte linear programming method. For several dimensions it yields improved upper bounds for kissing numbers and for spherical codes. Musin's recent work on kissing numbers in dimensions three and four can be viewed in our framework., 16 pages, 3 figures. Substantial changes after referee's comments, one new lemma