Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userOn Edge-Colored Saturation Problems
66
0
0.0
(
0
)
Ask ChatGPT about the research
No Arabic abstract
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 $rgeq binom{k-1}{2}+2$. 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.
rate research
Read More
We look at several saturation problems in complete balanced blow-ups of graphs. We let $H[n]$ denote the blow-up of $H$ onto parts of size $n$ and refer to a copy of $H$ in $H[n]$ as partite if it has one vertex in each part of $H[n]$. We then ask how few edges a subgraph $G$ of $H[n]$ can have such that $G$ has no partite copy of $H$ but such that the addition of any new edge from $H[n]$ creates a partite $H$. When $H$ is a triangle this value was determined by Ferrara, Jacobson, Pfender, and Wenger. Our main result is to calculate this value for $H=K_4$ when $n$ is large. We also give exact results for paths and stars and show that for $2$-connected graphs the answer is linear in $n$ whilst for graphs which are not $2$-connected the answer is quadratic in $n$. We also investigate a similar problem where $G$ is permitted to contain partite copies of $H$ but we require that the addition of any new edge from $H[n]$ creates an extra partite copy of $H$. This problem turns out to be much simpler and we attain exact answers for all cliques and trees.
It is conjectured that every edge-colored complete graph $G$ on $n$ vertices satisfying $Delta^{mon}(G)leq n-3k+1$ contains $k$ vertex-disjoint properly edge-colored cycles. We confirm this conjecture for $k=2$, prove several additional weaker results for general $k$, and we establish structural properties of possible minimum counterexamples to the conjecture. We also reveal a close relationship between properly edge-colored cycles in edge-colored complete graphs and directed cycles in multi-partite tournaments. Using this relationship and our results on edge-colored complete graphs, we obtain several partial solutions to a conjecture on disjoint cycles in directed graphs due to Bermond and Thomassen.
Properly colored cycles in edge-colored graphs are closely related to directed cycles in oriented graphs. As an analogy of the well-known Caccetta-H{a}ggkvist Conjecture, we study the existence of properly colored cycles of bounded length in an edge-colored graph. We first prove that for all integers $s$ and $t$ with $tgeq sgeq2$, every edge-colored graph $G$ with no properly colored $K_{s,t}$ contains a spanning subgraph $H$ which admits an orientation $D$ such that every directed cycle in $D$ is a properly colored cycle in $G$. Using this result, we show that for $rgeq4$, if the Caccetta-H{a}ggkvist Conjecture holds , then every edge-colored graph of order $n$ with minimum color degree at least $n/r+2sqrt{n}+1$ contains a properly colored cycle of length at most $r$. In addition, we also obtain an asymptotically tight total color degree condition which ensures a properly colored (or rainbow) $K_{s,t}$.
Let $G = (V, E)$ be an $n$-vertex edge-colored graph. In 2013, H. Li proved that if every vertex $v in V$ is incident to at least $(n+1)/2$ distinctly colored edges, then $G$ admits a rainbow triangle. We prove that the same hypothesis ensures a rainbow $ell$-cycle $C_{ell}$ whenever $n ge 432 ell$. This result is sharp for all odd integers $ell geq 3$, and extends earlier work of the authors for when $ell$ is even.
Let $G$ be a graph of order $n$ with an edge-coloring $c$, and let $delta^c(G)$ denote the minimum color degree of $G$. A subgraph $F$ of $G$ is called rainbow if all edges of $F$ have pairwise distinct colors. There have been a lot results on rainbow cycles of edge-colored graphs. In this paper, we show that (i) if $delta^c(G)>frac{3n-3}{4}$, then every vertex of $G$ is contained in a rainbow triangle; (ii) $delta^c(G)>frac{3n}{4}$, then every vertex of $G$ is contained in a rainbow $C_4$; and (iii) if $G$ is complete, $ngeq 8k-18$ and $delta^c(G)>frac{n-1}{2}+k$, then $G$ contains a rainbow cycle of length at least $k$. Some gaps in previous publications are also found and corrected.
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
