Related papers: Graphs with bounded tree-width and large odd-girth…
By the Grid Minor Theorem of Robertson and Seymour, every graph of sufficiently large tree-width contains a large grid as a minor. Tree-width may therefore be regarded as a measure of 'grid-likeness' of a graph. The grid contains a long…
A proper edge coloring of a graph $G$ with colors $1,2,\dots,t$ is called a \emph{cyclic interval $t$-coloring} if for each vertex $v$ of $G$ the edges incident to $v$ are colored by consecutive colors, under the condition that color $1$ is…
Mixed graphs can be seen as digraphs with arcs and edges (or digons, that is, two opposite arcs). In this paper, we consider the case where such graphs are bipartite and in which the undirected and directed degrees are one. The best graphs,…
We study minimum degree conditions for which a graph with given odd girth has a simple structure. For example, the classical work of Andr\'asfai, Erd\H os, and S\'os implies that every $n$-vertex graph with odd girth $2k+1$ and minimum…
Reed conjectured that for every $\varepsilon>0$ and every integer $\Delta$, there exists $g$ such that the fractional total chromatic number of every graph with maximum degree $\Delta$ and girth at least $g$ is at most…
Luo, Tian and Wu conjectured in 2022 that for any tree $T$ with bipartition $X$ and $Y$, every $k$-connected bipartite graph $G$ with $\delta(G) \geq k + t$, where $t = \max\{|X|,|Y |\}$, contains a subtree $T' \cong T$ such that $G-V(T')$…
Given a signed bipartite graph $(B, \pi)$ of negative girth $2k$, we present a necessary and sufficient condition for it to have the following property: each signed bipartite graph $(G, \sigma)$ whose negative girth is at least $2k$ and…
Concerning the recent notion of circular chromatic number of signed graphs, for each given integer $k$ we introduce two signed bipartite graphs, each on $2k^2-k+1$ vertices, having shortest negative cycle of length $2k$, and the circular…
Erd\H{o}s proved that there are graphs with arbitrarily large girth and chromatic number. We study the extension of this for generalized chromatic numbers.
For an odd integer $k$, let $\mathcal{C}_k = \{C_3,C_5,...,C_k\}$ denote the family of all odd cycles of length at most $k$ and let $\mathcal{C}$ denote the family of all odd cycles. Erd\H{o}s and Simonovits \cite{ESi1} conjectured that for…
In this paper, we investigate odd graceful graph, odd strongly harmonious graph, bipartite graph and their relationship. We proved following results: (1) if G is odd strongly harmonious graph, then G is odd graceful graph ;(2) if G is…
K\"onig's edge coloring theorem says that a bipartite graph with maximal degree $n$ has an edge coloring with no more than $n$ colors. We explore the computability theory and Reverse Mathematics aspects of this theorem. Computable bipartite…
We prove that there is a constant $c >0$, such that whenever $p \ge n^{-c}$, with probability tending to 1 when $n$ goes to infinity, every maximum triangle-free subgraph of the random graph $G_{n,p}$ is bipartite. This answers a question…
We determine the colored patterns that appear in any $2$-edge coloring of $K_{n,n}$, with $n$ large enough and with sufficient edges in each color. We prove the existence of a positive integer $z_2$ such that any $2$-edge coloring of…
Erd\"os conjectured that if $G$ is a triangle free graph of chromatic number at least $k\geq 3$, then it contains an odd cycle of length at least $k^{2-o(1)}$ \cite{sudakovverstraete, verstraete}. Nothing better than a linear bound…
Alon and Krivelevich conjectured that if $G$ is a bipartite graph of maximum degree $\Delta$, then the choosability (or list chromatic number) of $G$ satisfies $\chi_{\ell}(G) = O \left ( \log \Delta \right )$. Currently, the best known…
It is well-known that an undirected graph has no odd cycle if and only if it is bipartite. A less obvious, but similar result holds for directed graphs: a strongly connected digraph has no odd cycle if and only if it is bipartite. Can this…
A b-coloring of a graph is a coloring of its vertices such that every color class contains a vertex that has a neighbor in all other classes. The b-chromatic number of a graph is the largest integer k such that the graph has a b-coloring…
It is known (Bollob\'{a}s (1978); Kostochka and Mazurova (1977)) that there exist graphs of maximum degree $\Delta$ and of arbitrarily large girth whose chromatic number is at least $c \Delta / \log \Delta$. We show an analogous result for…
Birmele [J. Graph Theory, 2003] proved that every graph with circumference t has treewidth at most t-1. Under the additional assumption of 2-connectivity, such graphs have bounded pathwidth, which is a qualitatively stronger result.…