Related papers: Hamiltonicity of Bell and Stirling Colour Graphs

For a positive integer $k$ and a graph $H$ on $k$ vertices, we are interested in the inducibility of $H$, denoted $\mathrm{ind}(H)$, which is defined as the maximum possible probability that choosing $k$ vertices uniformly at random from a…

Call a colouring of a graph distinguishing, if the only colour preserving automorphism is the identity. A conjecture of Tucker states that if every automorphism of a graph $G$ moves infinitely many vertices, then there is a distinguishing…

We introduce a notion of color-criticality in the context of chromatic-choosability. We define a graph $G$ to be strong $k$-chromatic-choosable if $\chi(G) = k$ and every $(k-1)$-assignment for which $G$ is not list-colorable has the…

A graph is k-total colourable if there is an assignment of k different colours to the vertices and edges of the graph such that no two adjacent nor incident elements receive the same colour. The total chromatic number of some direct product…

If the vertices of a graph $G$ are colored with $k$ colors such that no adjacent vertices receive the same color and the sizes of any two color classes differ by at most one, then $G$ is said to be equitably $k$-colorable. Let $|G|$ denote…

A fall $k$-coloring of a graph $G$ is a proper $k$-coloring of $G$ such that each vertex of $G$ sees all $k$ colors on its closed neighborhood. We denote ${\rm Fall}(G)$ the set of all positive integers $k$ for which $G$ has a fall…

A $k$-colouring (not necessarily proper) of vertices of a graph is called {\it acyclic}, if for every pair of distinct colours $i$ and $j$ the subgraph induced by the edges whose endpoints have colours $i$ and $j$ is acyclic. In the paper…

We say that an edge colouring breaks an automorphism if some edge is mapped to an edge of a different colour. We say that the colouring is distinguishing if it breaks every non-identity automorphism. We show that such colouring can be…

Chv\'{a}tal conjectured in 1973 the existence of some constant $t$ such that all $t$-tough graphs with at least three vertices are hamiltonian. While the conjecture has been proven for some special classes of graphs, it remains open in…

Given two graphs $G$ and $H$, the $k$-colored Gallai-Ramsey number $gr_k(G : H)$ is defined to be the minimum integer $n$ such that every $k$-coloring of the complete graph on $n$ vertices contains either a rainbow copy of $G$ or a…

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…

In this paper uniquely list colorable graphs are studied. A graph G is called to be uniquely k-list colorable if it admits a k-list assignment from which G has a unique list coloring. The minimum k for which G is not uniquely k-list…

A signed tree-coloring of a signed graph $(G,\sigma)$ is a vertex coloring $c$ so that $G^{c}(i,\pm)$ is a forest for every $i\in c(u)$ and $u\in V(G)$, where $G^{c}(i,\pm)$ is the subgraph of $(G,\sigma)$ whose vertex set is the set of…

For given graph $H$ and graphical property $P$, the conditional chromatic number $\chi(H,P)$ of $H$, is the smallest number $k$, so that $V(H)$ can be decomposed into sets $V_1,V_2,\ldots, V_k$, in which $H[V_i]$ satisfies the property $P$,…

A balanced colouring of a graph is one in which every colour appears the same number of times. Given a fixed graph $H$ on $r$ vertices and a balanced $k$-colouring of the complete graph $K_{nrk}$, Hollom (2025) asked the following question:…

A graph $G$ is said to be $k$-distinguishable if the vertex set can be colored using $k$ colors such that no non-trivial automorphism fixes every color class, and the distinguishing number $D(G)$ is the least integer $k$ for which $G$ is…

A k-dimensional box is the Cartesian product R_1 x R_2 x ... x R_k where each R_i is a closed interval on the real line. The boxicity of a graph G, denoted as box(G) is the minimum integer k such that G is the intersection graph of a…

For graphs $G$ and $H$, an $H$-colouring of $G$ is a map $\psi:V(G)\rightarrow V(H)$ such that $ij\in E(G)\Rightarrow\psi(i)\psi(j)\in E(H)$. The number of $H$-colourings of $G$ is denoted by $\hom(G,H)$. We prove the following: for all…

For a fixed number of colors, we show that, in node-weighted split graphs, cographs, and graphs of bounded tree-width, one can determine in polynomial time whether a proper list-coloring of the vertices of a graph such that the total weight…

A graph is $(c_1, c_2, ..., c_k)$-colorable if the vertex set can be partitioned into $k$ sets $V_1,V_2, ..., V_k$, such that for every $i: 1\leq i\leq k$ the subgraph $G[V_i]$ has maximum degree at most $c_i$. We show that every planar…