Related papers: Locally self-avoiding eulerian tours

Tuza famously conjectured in 1981 that in a graph without k+1 edge-disjoint triangles, it suffices to delete at most 2k edges to obtain a triangle-free graph. The conjecture holds for graphs with small treewidth or small maximum average…

In 2018, by Ramsey and Hoffman theory, Koolen, Yang, and Yang presented a structural result on graphs with smallest eigenvalue at least $-3$ and large minimum degree. In this study, we depart from the conventional use of Ramsey theory and…

Thomassen's chord conjecture from 1976 states that every longest cycle in a $3$-connected graph has a chord. The circumference $c(G)$ and induced circumference $c'(G)$ of a graph $G$ are the length of its longest cycles and the length of…

The famous List Colouring Conjecture from the 1970s states that for every graph $G$ the chromatic index of $G$ is equal to its list chromatic index. In 1996 in a seminal paper, Kahn proved that the List Colouring Conjecture holds…

In 1994, Thomassen famously proved that every planar graph is 5-choosable, resolving a conjecture initially posed by Vizing and, independently, Erd\H{os}, Rubin, and Taylor in the 1970s. Later, Thomassen proved that every planar graph of…

The 1-2-3 Conjecture, introduced by Karo\'nski, {\L}uczak, and Thomason in 2004, was recently solved by Keusch. This implies that, for any connected graph $G$ different from $K_2$, we can turn $G$ into a locally irregular multigraph $M(G)$,…

We study minimum degree conditions under which a graph $G$ contains $k$th powers of paths and cycles of arbitrary specified lengths. We determine precise thresholds, assuming that the order of $G$ is large. This extends a result of Allen,…

We show that the edges of any $d$-regular graph can be almost decomposed into paths of length roughly $d$, giving an approximate solution to a problem of Kotzig from 1957. Along the way, we show that almost all of the vertices of a…

Much of extremal graph theory has concentrated either on finding very small subgraphs of a large graph (Turan-type results) or on finding spanning subgraphs (Dirac-type results). In this paper we are interested in finding intermediate-sized…

Recently, Berge theta hypergraphs have received special attention due to the similarity with Berge even cycles. Let $r$-uniform Berge theta hypergraph $\Theta_{\ell,t}^{B}$ be the $r$-uniform hypergraph consisting of $t$ internally disjoint…

We prove a precise min-max theorem for the following problem. Let $G$ be an Eulerian graph with a specified set of edges $S \subseteq E(G)$, and let $b$ be a vertex of $G$. Then what is the maximum integer $k$ so that the edge-set of $G$…

An oriented graph $H$ is Tur\'anable (resp. tileable) if there exist $n_0 \in \mathbb{N}$ such that every semi-regular near-tournament on $n \ge n_0$ vertices contains a copy of $H$ (resp. a perfect $H$-tiling). We disprove a conjectured…

A graph is $P_8$-free if it contains no induced subgraph isomorphic to the path $P_8$ on eight vertices. In 1995, Erd\H{o}s and Gy\'{a}rf\'{a}s conjectured that every graph of minimum degree at least three contains a cycle whose length is a…

Luo, Tian and Wu [Discrete Math. 345 (4) (2022) 112788] conjectured that for any tree $T$ with bipartition $(X,Y)$, every $k$-connected bipartite graph $G$ with minimum degree at least $k+w$, where $w=\max\{|X|,|Y|\}$, contains a tree…

Recently L. B. Beasley introduced $(2,3)$-cordial labelings of directed graphs in [1]. He made two conjectures which we resolve in this article. He conjectured that every orientation of a path of length at least five is $(2,3)$ cordial, and…

In 1979 Frankl conjectured that in a finite non-trivial union-closed collection of sets there has to be an element that belongs to at least half the sets. We show that this is equivalent to the conjecture that in a finite non-trivial graph…

B. Bollob\'{a}s and G. Brightwell and independently R. Shi proved the existence of a cycle through all vertices whose degrees at least $\frac{n}{2}$ in any $2$-connected graph of order $n$. Motivated by this result, we prove the existence…

We prove that if $T_1,\dots, T_n$ is a sequence of bounded degree trees so that $T_i$ has $i$ vertices, then $K_n$ has a decomposition into $T_1,\dots, T_n$. This shows that the tree packing conjecture of Gy\'arf\'as and Lehel from 1976…

In a sequence of four papers, we prove the following results (via a unified approach) for all sufficiently large $n$: (i) [1-factorization conjecture] Suppose that $n$ is even and $D\geq 2\lceil n/4\rceil -1$. Then every $D$-regular graph…

In an early work from 1896, Maschke established the complete list of all finite planar Cayley graphs. This result initiated a long line of research over the next century, aiming at characterizing in a similar way all planar infinite Cayley…