Related papers: Decomposition of planar graphs with forbidden conf…

We show that for each \ell\geq 4 every sufficiently large oriented graph G with \delta^+(G), \delta^-(G) \geq \lfloor |G|/3 \rfloor +1 contains an \ell-cycle. This is best possible for all those \ell\geq 4 which are not divisible by 3.…

It was conjectured by Steinberg in 1976 that planar graphs without cycles of length 4 or 5 are 3-colorable. This conjecture attracted a substantial amount of attention and was finally refuted by Cohen-Addad, Hebdige, Kr\'{a}l', Li and…

Gallai's path decomposition conjecture states that if $G$ is a connected graph on $n$ vertices, then the edges of $G$ can be decomposed into at most $\lceil \frac{n }{2} \rceil$ paths. A graph is said to be an odd semi-clique if it can be…

Let $G$ be a planar graph with no two 3-cycles sharing an edge. We show that if $\Delta(G)\geq 9$, then $\chi'_l(G) = \Delta(G)$ and $\chi''_l(G)=\Delta(G)+1.$ We also show that if $\Delta(G)\geq 6$, then $\chi'_l(G)\leq\Delta(G)+1$ and if…

A graph $G$ is $(I,F)$-partitionable if its vertex set can be partitioned into two parts such that one part is an independent set, and the other induces a forest. In this paper, we prove that every planar graph without cycles of length $4,…

A graph $G$ has a $C_k$-decomposition if its edge set can be partitioned into cycles of length $k$. We show that if $\delta(G)\geq 2|G|/3-1$, then $G$ has a $C_4$-decomposition, and if $\delta(G)\geq |G|/2$, then $G$ has a…

In this work, we study conditions for the existence of length-constrained path-cycle decompositions, that is, partitions of the edge set of a graph into paths and cycles of a given minimum length. Our main contribution is the…

As usual, $P_n$ ($n \geq 1$) denotes the path on $n$ vertices, and $C_n$ ($n \geq 3$) denotes the cycle on $n$ vertices. For a family $\mathcal{H}$ of graphs, we say that a graph $G$ is $\mathcal{H}$-free if no induced subgraph of $G$ is…

A graph is $(d_1, ..., d_r)$-colorable if its vertex set can be partitioned into $r$ sets $V_1, ..., V_r$ so that the maximum degree of the graph induced by $V_i$ is at most $d_i$ for each $i\in \{1, ..., r\}$. For a given pair $(g, d_1)$,…

An odd graph is a finite graph all of whose vertices have odd degrees. Given graph $G$ is decomposable into $k$ odd subgraphs if its edge set can be partitioned into $k$ subsets each of which induces an odd subgraph of $G$. The minimum…

A list assignment $L$ for a graph $G$ is an $(\ell,k)$-list assignment if $|L(v)|\geq \ell$ for each $v \in V(G)$ and $|L(u) \cap L(v)| \leq k$ for each $uv \in E(G)$. We say $G$ is $(\ell,k)$-choosable if it admits an $L$-colouring for…

A strong edge-coloring of a graph $G$ is a coloring of edges of $G$ such that every color class forms an induced matching. The strong chromatic index is the minimum number of colors needed to color the graph. The Ore-degree $\theta(G)$ of a…

An ordered graph is a pair $\mathcal{G}=(G,\prec)$ where $G$ is a graph and $\prec$ is a total ordering of its vertices. The ordered Ramsey number $\overline{R}(\mathcal{G})$ is the minimum number $N$ such that every $2$-coloring of the…

Montassier, Raspaud, and Wang (2006) asked to find the smallest positive integers $d_0$ and $d_1$ such that planar graphs without $\{4,5\}$-cycles and $d^{\Delta}\ge d_0$ are $3$-choosable and planar graphs without $\{4,5,6\}$-cycles and…

A graph is locally irregular if any pair of adjacent vertices have distinct degrees. A locally irregular decomposition of a graph $G$ is a decomposition $\mathcal{D}$ of $G$ such that every subgraph $H \in \mathcal{D}$ is locally irregular.…

Cai et al.\ proved that a toroidal graph $G$ without $6$-cycles is $5$-choosable, and proposed the conjecture that $\textsf{ch}(G) = 5$ if and only if $G$ contains a $K_{5}$ [J. Graph Theory 65 (2010) 1--15], where $\textsf{ch}(G)$ is the…

Odd coloring is a proper coloring with an additional restriction that every non-isolated vertex has some color that appears an odd number of times in its neighborhood. The minimum number of colors $k$ that can ensure an odd coloring of a…

A generalization of list-coloring, now known as DP-coloring, was recently introduced by Dvo\v{r}\'{a}k and Postle. Essentially, DP-coloring assigns an arbitrary matching between lists of colors at adjacent vertices, as opposed to only…

We study the problem of determining whether a given graph~$G=(V,E)$ admits a matching~$M$ whose removal destroys all odd cycles of~$G$ (or equivalently whether~$G-M$ is bipartite). This problem is equivalent to determine whether~$G$ admits…

A $P_\ell$-decomposition of a graph $G$ is a set of paths with $\ell$ edges in $G$ that cover the edge set of $G$. Favaron, Genest, and Kouider (2010) conjectured that every $(2k+1)$-regular graph that contains a perfect matching admits a…