Samal, Robert

Star chromatic index of complete graphs ★★

Author(s): Dvorak; Mohar; Samal

\begin{conjecture} Is it possible to color edges of the complete graph $K_n$ using $O(n)$ colors, so that the coloring is proper and no 4-cycle and no 4-edge path is using only two colors?

Equivalently: is the star chromatic index of $K_n$ linear in $n$? \end{conjecture}

Keywords: complete graph; edge coloring; star coloring

Star chromatic index of cubic graphs ★★

Author(s): Dvorak; Mohar; Samal

The star chromatic index $\chi_s'(G)$ of a graph $G$ is the minimum number of colors needed to properly color the edges of the graph so that no path or cycle of length four is bi-colored.

\begin{question} Is it true that for every (sub)cubic graph $G$, we have $\chi_s'(G) \le 6$? \end{question}

Keywords: edge coloring; star coloring

Weak pentagon problem ★★

Author(s): Samal

\begin{conjecture} If $G$ is a cubic graph not containing a triangle, then it is possible to color the edges of $G$ by five colors, so that the complement of every color class is a bipartite graph. \end{conjecture}

Keywords: Clebsch graph; cut-continuous mapping; edge-coloring; homomorphism; pentagon

Drawing disconnected graphs on surfaces ★★

Author(s): DeVos; Mohar; Samal

\begin{conjecture} Let $G$ be the disjoint union of the graphs $G_1$ and $G_2$ and let $\Sigma$ be a surface. Is it true that every optimal drawing of $G$ on $\Sigma$ has the property that $G_1$ and $G_2$ are disjoint? \end{conjecture}

Keywords: crossing number; surface

Cores of Cayley graphs ★★

Author(s): Samal

\begin{conjecture} Let $M$ be an abelian group. Is the \Def[core]{core (graph theory)} of a \Def{Cayley graph} (on some power of $M$) a Cayley graph (on some power of $M$)? \end{conjecture}

Keywords: Cayley graph; core

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