Properly coloured Hamiltonian cycles in edge-coloured complete graphs

Let Kc n be an edge-coloured complete graph on n vertices. Let Δmon(Kc n) denote the largest number of edges of the same colour incident with a vertex of Kc n. A properly coloured cycle is a cycle such that no two adjacent edges have the same colour. In 1976, BollobÁs and ErdŐs[6] conjectured that every Kc n with Δmon(Kc n)<⌊n/2⌋contains a properly coloured Hamiltonian cycle. In this paper, we show that for any ε>0, there exists an integer n0 such that every Kc n with Δmon(Kc n)<(1/2–ε)n and n≥n0 contains a properly coloured Hamiltonian cycle. This improves a result of Alon and Gutin [1]. Hence, the conjecture of BollobÁs and ErdŐs is true asymptotically

Alternating Hamiltonian cycles in 22-edge-colored multigraphs

A path (cycle) in a $2$-edge-colored multigraph is alternating if no two consecutive edges have the same color. The problem of determining the existence of alternating Hamiltonian paths and cycles in $2$-edge-colored multigraphs is an $\mathcal{NP}$-complete problem and it has been studied by several authors. In Bang-Jensen and Gutin's book "Digraphs: Theory, Algorithms and Applications", it is devoted one chapter to survey the last results on this topic. Most results on the existence of alternating Hamiltonian paths and cycles concern on complete and bipartite complete multigraphs and a few ones on multigraphs with high monochromatic degrees or regular monochromatic subgraphs. In this work, we use a different approach imposing local conditions on the multigraphs and it is worthwhile to notice that the class of multigraphs we deal with is much larger than, and includes, complete multigraphs, and we provide a full characterization of this class. Given a $2$-edge-colored multigraph $G$, we say that $G$ is $2$-$\mathcal{M}$-closed (resp. $2$-$\mathcal{NM}$-closed)} if for every monochromatic (resp. non-monochromatic) $2$-path $P=(x_1, x_2, x_3)$, there exists an edge between $x_1$ and $x_3$. In this work we provide the following characterization: A $2$-$\mathcal{M}$-closed multigraph has an alternating Hamiltonian cycle if and only if it is color-connected and it has an alternating cycle factor. Furthermore, we construct an infinite family of $2$-$\mathcal{NM}$-closed graphs, color-connected, with an alternating cycle factor, and with no alternating Hamiltonian cycle.Comment: 15 pages, 20 figure

A revisit to Bang-Jensen-Gutin conjecture and Yeo's theorem

A path (cycle) is properly-colored if consecutive edges are of distinct colors. In 1997, Bang-Jensen and Gutin conjectured a necessary and sufficient condition for the existence of a Hamilton path in an edge-colored complete graph. This conjecture, confirmed by Feng, Giesen, Guo, Gutin, Jensen and Rafley in 2006, was laterly playing an important role in Lo's asymptotical proof of Bollob\'as-Erd\H{o}s' conjecture on properly-colored Hamilton cycles. In 1997, Yeo obtained a structural characterization of edge-colored graphs that containing no properly colored cycles. This result is a fundamental tool in the study of edge-colored graphs. In this paper, we first give a much shorter proof of the Bang-Jensen-Gutin Conjecture by two novel absorbing lemmas. We also prove a new sufficient condition for the existence of a properly-colored cycle and then deduce Yeo's theorem from this result and a closure concept in edge-colored graphs.Comment: 13 pages, 5 figure

Properly Edge-colored Theta Graphs in Edge-colored Complete Graphs

With respect to specific cycle-related problems, edge-colored graphs can be considered as a generalization of directed graphs. We show that properly edge-colored theta graphs play a key role in characterizing the difference between edge-colored complete graphs and multipartite tournaments. We also establish sufficient conditions for an edge-colored complete graph to contain a small and a large properly edge-colored theta graph, respectively

Graph tilings in incompatibility systems

Given two graphs $H$ and $G$, an \emph{$H$-tiling} of $G$ is a collection of vertex-disjoint copies of $H$ in $G$ and an \emph{$H$-factor} is an $H$-tiling that covers all vertices of $G$. K\"{u}hn and Osthus managed to characterize, up to an additive constant, the minimum degree threshold which forces an $H$-factor in a host graph $G$. In this paper we study a similar tiling problem in a system that is locally bounded. An \emph{incompatibility system} $\mathcal{F}$ over $G$ is a family $\mathcal{F}=\{F_v\}_{v\in V(G)}$ with $F_v\subseteq \{\{e,e'\}\in {E(G)\choose 2}: e\cap e'=\{v\}\}$. We say that two edges $e,e'\in E(G)$ are \emph{incompatible} if $\{e,e'\}\in F_v$ for some $v\in V(G)$, and otherwise \emph{compatible}. A subgraph $H$ of $G$ is \emph{compatible} if every pair of edges in $H$ are compatible. An incompatibility system $\mathcal{F}$ is \emph{$\Delta$-bounded} if for any vertex $v$ and any edge $e$ incident with $v$, there are at most $\Delta$ two-subsets in $F_v$ containing $e$. This notion was partly motivated by a concept of transition system introduced by Kotzig in 1968, and first formulated by Krivelevich, Lee and Sudakov to study the robustness of Hamiltonicity of Dirac graphs. We prove that for any $\alpha>0$ and any graph $H$ with $h$ vertices, there exists a constant $\mu>0$ such that for any sufficiently large $n$ with $n\in h\mathbb{N}$, if $G$ is an $n$-vertex graph with $\delta(G)\ge(1-\frac{1}{\chi^*(H)}+\alpha)n$ and $\mathcal{F}$ is a $\mu n$-bounded incompatibility system over $G$, then there exists a compatible $H$-factor in $G$, where the value $\chi^*(H)$ is either the chromatic number $\chi(H)$ or the critical chromatic number $\chi_{cr}(H)$ and we provide a dichotomy. Moreover, the error term $\alpha n$ is inevitable in general case