The vertex arboricity $\mathrm{va}(G)$ of a multigraph $G$ is the minimum number $k$ for which $V(G)$ can be partitioned into $k$ subsets, each of which induces an acyclic subgraph of $G$. By definition, if $\mathrm{va}(G)= k$, then the chromatic number, $\chi(G)$, satisfies $k\leq \chi(G)\leq 2k$. Fundamental results by Borodin from 1976 and Bollob\'as and Manvel from 1979 imply an analog of Gallai's lower bound on the number of edges in a $(2k-1)$-critical graph. We consider a slight generalization of vertex arboricity in the setting of DP-coloring. Using this framework, we derive lower bounds on the number of edges in graphs critical for vertex arboricity and for list arboricity that are better than Gallai's bound, along with similar bounds in our DP-setting.
For an integer $k\geq2$, let $\chi_k'(G)$ denote the minimum number of colors in an edge-coloring of a graph $G$ such that every nonzero degree in each color subgraph is congruent to $1\pmod{k}$. A graph is a $0_k$-graph if every vertex degree is divisible by $k$. We disprove a conjecture of Berthe et al.\ (On modular...
A full-homomorphism from a graph $G$ to a graph $H$ is a function on vertex sets that preserves adjacency and non-adjacency of vertices. A graph $G$ is called a minimal $H$-obstruction if it has no full-homomorphism to $H$ but every proper vertex induced subgraph of $G$ does. Such graphs can have at most $|V(H)|+1$ ver...
Z. Rahimi, M. H. Shirdareh-Haghighi, Asma Namazi Department of Mathematics et al.· 0 citations
A dominating set in a graph $G$ is a subset $S$ of its vertices such that each vertex in $G$ is either in $S$ or adjacent to a vertex in $S$. Nordhaus-Gaddum inequalities relate the values of a graph parameter on a graph and its complement. In this setting, Keough and Shane conjecture that any graph $G$ on $n$ vertices...
For finite simple graphs $G,H$ on a common vertex set $V$, we say that $H$ is $G$-colorable if $H$ admits a proper list coloring with list assignment $L(v)=N_G(v)$ for all $v\in V$. This notion of coloring a graph using the neighborhood of another graph on the same vertex set, which we call \emph{graphic list coloring}...
The bounds on the number of Eulerian orientations for certain classes of connected, loopless $4-regular graphs are improved and a divide-and-conquer algorithm is provided that leverages structural properties to compute the exact number of Eulerian orientations for separable graphs without exhaustive enumeration.
A vertex cover of a graph $G$ is a subset of vertices $C \subseteq V(G)$ such that every edge of $G$ is incident to at least one vertex in $C$. The vertex cover number of $G$ is the minimum cardinality of a vertex cover of $G$ and is denoted by $\tau(G)$. A snark is a connected, bridgeless, cubic graph that has an edge...
G. Fernandes, A. Luiz· 0 citations
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