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Definition. Suppose $H=(V,E)$ is a hypergraph. Call a hyperedge $e=\{v_1,v_2,\dotsc,v_k\}$ a $k$-star if in the 1-skeleton of $H$ there is a copy of $K^{1,k-1}$ whose union equals $e$. (Obviously in t …

Let $G$ be a simple graph such that some of its vertices are like a fork. i.e. there is vertices $w,x,a,v$ such that edges $[v,w]$ and $[w,a]$ are incident in $w$ and edges $[w,a]$ and $[x,w]$ are …

This question is related to my previous post: New edge coloring problem in graph theory. Added: Let $G$ be a simple graph. Consider the following edge coloring: We are allowed to use repetitive co …

Let $G$ be a simple graph. Consider the following edge coloring: We are allowed to use repetitive colors on some edges incident to a vertex such that the result does not contain a sequence of length …

I have a question on the Total Coloring Conjecture in graph theory. This conjecture states that $$\chi^"(G)\leq \Delta +2,$$ where $\Delta$ is the maximum degree of the graph and $\chi^"(G)$ denotes …

Let $G$ be a simple graph such that $\omega(G)\leq\lfloor\frac{\Delta(G)+1}{2}\rfloor+1$ where $\Delta(G)$ is the maximal degree of $G$. Is it true that \begin{equation} \chi(G)\leq \lfloor\frac{\De …

Let $G$ be a simple graph with $n$ vertices. Let $\omega(G)$ and $\chi(G)$ denotes the clique number and chromatic number of $G$ respectively. Then Does $\omega(G)\leq k$ imply $\chi(G)\leq k$ for …

I want to know whether there are ways to use algebraic methods for solving graph theory problems (graph coloring problems). For example, is it possible to prove the four-color theorem purely with alg …

I am a beginner in graph theory and I am interested in finding an upper bound for the chromatic number of the following class of graphs: If two vertices $a$ and $b$ are adjacent in $G$, then th …