No Arabic abstract
Barnette identified two interesting classes of cubic polyhedral graphs for which he conjectured the existence of a Hamiltonian cycle. Goodey proved the conjecture for the intersection of the two classes. We examine these classes from the point of view of distance-two colorings. A distance-two $r$-coloring of a graph $G$ is an assignment of $r$ colors to the vertices of $G$ so that any two vertices at distance at most two have different colors. Note that a cubic graph needs at least four colors. The distance-two four-coloring problem for cubic planar graphs is known to be NP-complete. We claim the problem remains NP-complete for tri-connected bipartite cubic planar graphs, which we call type-one Barnette graphs, since they are the first class identified by Barnette. By contrast, we claim the problem is polynomial for cubic plane graphs with face sizes $3, 4, 5,$ or $6$, which we call type-two Barnette graphs, because of their relation to Barnettes second conjecture. We call Goodey graphs those type-two Barnette graphs all of whose faces have size $4$ or $6$. We fully describe all Goodey graphs that admit a distance-two four-coloring, and characterize the remaining type-two Barnette graphs that admit a distance-two four-coloring according to their face size. For quartic plane graphs, the analogue of type-two Barnette graphs are graphs with face sizes $3$ or $4$. For this class, the distance-two four-coloring problem is also polynomial; in fact, we can again fully describe all colorable instances -- there are exactly two such graphs.
$H_q(n,d)$ is defined as the graph with vertex set ${mathbb Z}_q^n$ and where two vertices are adjacent if their Hamming distance is at least $d$. The chromatic number of these graphs is presented for various sets of parameters $(q,n,d)$. For the $4$-colorings of the graphs $H_2(n,n-1)$ a notion of robustness is introduced. It is based on the tolerance of swapping colors along an edge without destroying properness of the coloring. An explicit description of the maximally robust $4$-colorings of $H_2(n,n-1)$ is presented.
A graph drawn in the plane with n vertices is k-fan-crossing free for k > 1 if there are no k+1 edges $g,e_1,...e_k$, such that $e_1,e_2,...e_k$ have a common endpoint and $g$ crosses all $e_i$. We prove a tight bound of 4n-8 on the maximum number of edges of a 2-fan-crossing free graph, and a tight 4n-9 bound for a straight-edge drawing. For k > 2, we prove an upper bound of 3(k-1)(n-2) edges. We also discuss generalizations to monotone graph properties.
A proper edge-coloring of a graph $G$ with colors $1,ldots,t$ is called an emph{interval cyclic $t$-coloring} if all colors are used, and the edges incident to each vertex $vin V(G)$ are colored by $d_{G}(v)$ consecutive colors modulo $t$, where $d_{G}(v)$ is the degree of a vertex $v$ in $G$. A graph $G$ is emph{interval cyclically colorable} if it has an interval cyclic $t$-coloring for some positive integer $t$. The set of all interval cyclically colorable graphs is denoted by $mathfrak{N}_{c}$. For a graph $Gin mathfrak{N}_{c}$, the least and the greatest values of $t$ for which it has an interval cyclic $t$-coloring are denoted by $w_{c}(G)$ and $W_{c}(G)$, respectively. In this paper we investigate some properties of interval cyclic colorings. In particular, we prove that if $G$ is a triangle-free graph with at least two vertices and $Gin mathfrak{N}_{c}$, then $W_{c}(G)leq vert V(G)vert +Delta(G)-2$. We also obtain bounds on $w_{c}(G)$ and $W_{c}(G)$ for various classes of graphs. Finally, we give some methods for constructing of interval cyclically non-colorable graphs.
An edge-coloring of a graph $G$ with consecutive integers $c_{1},ldots,c_{t}$ is called an emph{interval $t$-coloring} if all colors are used, and the colors of edges incident to any vertex of $G$ are distinct and form an interval of integers. A graph $G$ is interval colorable if it has an interval $t$-coloring for some positive integer $t$. The set of all interval colorable graphs is denoted by $mathfrak{N}$. In 2004, Giaro and Kubale showed that if $G,Hin mathfrak{N}$, then the Cartesian product of these graphs belongs to $mathfrak{N}$. In the same year they formulated a similar problem for the composition of graphs as an open problem. Later, in 2009, the first author showed that if $G,Hin mathfrak{N}$ and $H$ is a regular graph, then $G[H]in mathfrak{N}$. In this paper, we prove that if $Gin mathfrak{N}$ and $H$ has an interval coloring of a special type, then $G[H]in mathfrak{N}$. Moreover, we show that all regular graphs, complete bipartite graphs and trees have such a special interval coloring. In particular, this implies that if $Gin mathfrak{N}$ and $T$ is a tree, then $G[T]in mathfrak{N}$.
We show that any proper coloring of a Kneser graph $KG_{n,k}$ with $n-2k+2$ colors contains a trivial color (i.e., a color consisting of sets that all contain a fixed element), provided $n>(2+epsilon)k^2$, where $epsilonto 0$ as $kto infty$. This bound is essentially tight.