Modern Graph Theory

The rainbow Turan number $\mathrm{ex}^*(n,H)$ of a graph $H$ is the maximum possible number of edges in a properly edge-coloured $n$-vertex graph with no rainbow subgraph isomorphic to $H$. We prove that for any integer $k\geq 2$, $\mathrm{ex}^*(n,C_{2k})=O(n^{1+1/k})$. This is tight and establishes a conjecture of Keevash, Mubayi, Sudakov and Verstraete. We use the same method to prove several other conjectures in various topics. First, we prove that there exists a constant $c$ such that any properly edge-coloured $n$-vertex graph with more than $cn(\log n)^4$ edges contains a rainbow cycle. It is known that there exist properly edge-coloured $n$-vertex graphs with $\Omega(n\log n)$ edges which do not contain any rainbow cycle. Secondly, we show that in any proper edge-colouring of $K_n$ with $o(n^{\frac{r}{r-1}\cdot \frac{k-1}{k}})$ colours, there exist $r$ colour-isomorphic, pairwise vertex-disjoint copies of $C_{2k}$. This proves in a strong form a conjecture of Conlon and Tyomkyn, and a strenghtened version proposed by Xu, Zhang, Jing and Ge. Moreover, we answer a question of Jiang and Newman by showing that there exists a constant $c=c(r)$ such that any $n$-vertex graph with more than $cn^{2-1/r}(\log n)^{7/r}$ edges contains the $r$-blowup of an even cycle. Finally, we prove that the $r$-blowup of $C_{2k}$ has Turan number $O(n^{2-\frac{1}{r}+\frac{1}{k+r-1}+o(1)})$, which can be used to disprove an old conjecture of Erd\H os and Simonovits.

Rainbow Turán number of even cycles, repeated patterns and blow-ups of cycles

Alon and Shikhelman initiated the systematic study of the following generalized Turan problem: for fixed graphs $H$ and $F$ and an integer $n$, what is the maximum number of copies of $H$ in an $n$-vertex $F$-free graph? 
An edge-colored graph is called rainbow if all its edges have different colors. The rainbow Turan number of $F$ is defined as the maximum number of edges in a properly edge-colored graph on $n$ vertices with no rainbow copy of $F$. The study of rainbow Turan problems was initiated by Keevash, Mubayi, Sudakov and Verstraete. 
Motivated by the above problems, we study the following problem: What is the maximum number of copies of $F$ in a properly edge-colored graph on $n$ vertices without a rainbow copy of $F$? We establish several results, including when $F$ is a path, cycle or tree.

Generalized rainbow Tur\'an problems

In 1959 Erd\H os and Gallai proved the asymptotically optimal bound for the maximum number of edges in graphs not containing a path of a fixed length. We investigate a rainbow version of the theorem, in which one considers $k \geq 1$ graphs on a common set of vertices not creating a path having edges from different graphs and asks for the maximum number of edges in each graph. We prove the asymptotically optimal bound in the case of a path on three edges and any $k \geq 1$.

https://journals.muni.cz/eurocomb/article/download/35545/30281

Graphs without a rainbow path of length 3

Our first theorem in this paper is a hierarchy theorem for the query complexity of testing graph properties with 1-sided error; more precisely, we show that for every sufficiently fast-growing function f, there is a graph property whose 1-sided-error query complexity is precisely f(Θ(1/e)). No result of this type was previously known for any f which is super-polynomial. Goldreich [ECCC 2005] asked to exhibit a graph property whose query complexity is 2Θ(1/e). Our hierarchy theorem partially resolves this problem by exhibiting a property whose 1-sided-error query complexity is 2Θ(1/e). We also use our hierarchy theorem in order to resolve a problem raised by the second author and Alon [STOC 2005] regarding testing relaxed versions of bipartiteness. Our second theorem states that for any function f there is a graph property whose 1-sided-error query complexity is f(Θ(1/e)) while its 2-sided-error query complexity is only poly(1/e). This is the first indication of the surprising power that 2-sided-error testing algorithms have over 1-sided-error ones, even when restricted to properties that are testable with 1-sided error. Again, no result of this type was previously known for any f that is super polynomial. The above theorems are derived from a graph theoretic result which we think is of independent interest, and might have further applications. Alon and Shikhelman [JCTB 2016] introduced the following generalized Turan problem: for fixed graphs H and T, and an integer n, what is the maximum number of copies of T, denoted by ex(n,T,H), that can appear in an n-vertex H-free graph? This problem received a lot of attention recently, with an emphasis on ex(n,C3,C2l +1). Our third theorem in this paper gives tight bounds for ex(n,Ck,Cl) for all the remaining values of k and l.

A Generalized Turan Problem and its Applications.

For a fixed graph $F$, we would like to determine the maximum number of edges in a properly edge-colored graph on $n$ vertices which does not contain a rainbow copy of $F$, that is, a copy of $F$ all of whose edges receive a different color. This maximum, denoted by $ex^*(n, F)$, is the rainbow Turan number of $F$. We show that $ex^*(n,P_k)\geq \frac{k}{2}n + O(1)$ where $P_k$ is a path on $k\geq 3$ edges, generalizing a result by Maamoun and Meyniel and by Johnston, Palmer and Sarkar. We show similar bounds for brooms on $2^s-1$ edges and diameter $\leq 10$ and a few other caterpillars of small diameter.

Lower bounds for rainbow Tur\'{a}n numbers of paths and other trees

We introduce a notion of rainbow saturation and the corresponding rainbow saturation number. This is the saturation version of the rainbow Tur\'an numbers whose systematic study was initiated by Keevash, Mubayi, Sudakov, and Verstra\"ete. We give examples of graphs for which the rainbow saturation number is bounded away from the ordinary saturation number. This includes all complete graphs $K_n$ for $n\geq 4$, and several bipartite graphs. It is notable that there are non-bipartite graphs for which this is the case, as this does not happen when it comes to the rainbow extremal number versus the traditional extremal number. We also show that saturation numbers are linear for a large class of graphs, providing a partial rainbow analogue of a well known theorem of K\'asonyi and Tuza. We conclude this paper with related open questions and conjectures.

Rainbow Saturation

A graph $G$ is $d$-distinguishable if there is a coloring of the vertices with $d$ colors so that only the trivial automorphism preserves the color classes. The smallest such $d$ is the distinguishing number, $\operatorname{Dist}(G)$. The Mycielskian $\mu(G)$ of a graph $G$ is constructed by adding a shadow vertex $u_i$ for each vertex $v_i$ of $G$ and one additional vertex $w$ and adding edges so that $N(u_i)~=~N_G(v_i)~\cup~\{w\}$. The generalized Mycielskian $\mu^{(t)}(G)$ is a Mycielskian graph with $t$ layers of shadow vertices, each with edges to layers above and below. This paper examines the distinguishing number of the traditional and generalized Mycielskian graphs. Notably, if $G~\neq ~K_1,~K_2$ and the number of isolated vertices in $\mu^{(t)}(G)$ is at most $\operatorname{Dist}(G)$, then $\operatorname{Dist}(\mu^{(t)}(G)) \le \operatorname{Dist}(G)$. This result proves and exceeds a conjecture of Alikhani and Soltani.

Distinguishing Generalized Mycielskian Graphs

A set $S$ of vertices is a determining set for a graph $G$ if every automorphism of $G$ is uniquely determined by its action on $S$. The size of a smallest determining set for $G$ is called its determining number, $Det(G)$. A graph $G$ is said to be $d$-distinguishable if there is a coloring of the vertices with $d$ colors so that only the trivial automorphism preserves the color classes. The smallest such $d$ is the distinguishing number, $Dist(G)$. If $Dist(G) = 2$, the cost of 2-distinguishing, $\rho(G)$, is the size of a smallest color class over all 2-distinguishing colorings of $G$. The Mycielskian, $\mu(G)$, of a graph $G$ is constructed by adding a shadow master vertex $w$, and for each vertex $v_i$ of $G$ adding a shadow vertex $u_i$ with edges so that the neighborhood of $u_i$ in $\mu(G)$ is the same as the neighborhood of $v_i$ in $G$ with the addition of $w$. That is, $N(u_i)=N_G(v_i)\cup\{w\}$. The generalized Mycielskian $\mu^{(t)}(G)$ of a graph $G$ is a Mycielskian graph with $t$ layers of shadow vertices, each with edges to layers above and below, and $w$ only adjacent to the top layer of shadow vertices. A graph is twin-free if it has no pair of vertices with the same set of neighbors. This paper examines the determining number and, when relevant, the cost of 2-distinguishing for Mycielskians and generalized Mycielskians of simple graphs with no isolated vertices. In particular, if $G \neq K_2$ is twin-free with no isolated vertices, then $Det(\mu^{(t)}(G)) = Det(G)$. Further, if $Det(G) = k \geq 2$ and $t \ge k-1$, then $Dist(\mu^{(t)}(G))=2$, and $Det(\mu^{(t)}(G)) = \rho(\mu^{(t)}(G))= k$. For $G$ with twins, we develop a framework using quotient graphs with respect to equivalence classes of twin vertices to give bounds on the determining number of Mycielskians. Moreover, we identify classes of graphs with twins for which $Det(\mu^{(t)}(G)) = (t{+}1) Det(G)$.

pdf/determining-number-and-cost-of-generalized-mycielskian-19u4ahcftb.pdf

Determining Number and Cost of Generalized Mycielskian Graphs

The Mycielskian construction, denoted μ(G), takes a finite simple graph G to a larger graph with of the same clique number but larger chromatic number. The generalized Mycielskian construction, denoted μt(G), takes G to a larger graph with the same chromatic number but with larger odd girth. In this chapter we look at symmetry parameters of μ(G) and μt(G) in terms of the same parameters of G. These symmetry parameters include determining number, distinguishing number, and cost of distinguishing.

Symmetry Parameters for Mycielskian Graphs

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