Orthogonal Colourings of Cayley Graphs

Two colourings of a graph are orthogonal if they have the property that when two vertices are coloured with the same colour in one colouring, then those vertices receive distinct colours in the other colouring. In this paper, orthogonal colourings of Cayley graphs are discussed. Firstly, the orthogonal chromatic number of cycle graphs are completely determined. Secondly, the orthogonal chromatic number of certain circulant graphs is explored. Lastly, orthogonal colourings of product graphs and Hamming graphs are studied.Comment: 13 pages, 4 figure

A Fractional Analogue of Brooks' Theorem

Let $\Delta(G)$ be the maximum degree of a graph $G$. Brooks' theorem states that the only connected graphs with chromatic number $\chi(G)=\Delta(G)+1$ are complete graphs and odd cycles. We prove a fractional analogue of Brooks' theorem in this paper. Namely, we classify all connected graphs $G$ such that the fractional chromatic number $\chi_f(G)$ is at least $\Delta(G)$. These graphs are complete graphs, odd cycles, $C^2_8$, $C_5\boxtimes K_2$, and graphs whose clique number $\omega(G)$ equals the maximum degree $\Delta(G)$. Among the two sporadic graphs, the graph $C^2_8$ is the square graph of cycle $C_8$ while the other graph $C_5\boxtimes K_2$ is the strong product of $C_5$ and $K_2$. In fact, we prove a stronger result; if a connected graph $G$ with $\Delta(G)\geq 4$ is not one of the graphs listed above, then we have $\chi_f(G)\leq \Delta(G)- 2/67$.Comment: Third version, add Andrew King as an coautho

Strong edge coloring of Cayley graphs and some product graphs

A strong edge coloring of a graph $G$ is a proper edge coloring of $G$ such that every color class is an induced matching. The minimum number of colors required is termed the strong chromatic index. In this paper, we determine the exact value of the strong chromatic index of all unitary Cayley graphs. Our investigations reveal an underlying product structure from which the unitary Cayley graphs emerge. We then go on to give tight bounds for the strong chromatic index of the Cartesian product of two trees, including an exact formula for the product in the case of stars. Further, we give bounds for the strong chromatic index of the product of a tree with a cycle. For any tree, those bounds may differ from the actual value only by not more than a small additive constant (at most 2 for even cycles and at most 5 for odd cycles), moreover they yield the exact value when the length of the cycle is divisible by $4$

Coloring graphs using topology

Higher dimensional graphs can be used to colour two-dimensional geometric graphs. If G the boundary of a three dimensional graph H for example, we can refine the interior until it is colourable with 4 colours. The later goal is achieved if all interior edge degrees are even. Using a refinement process which cuts the interior along surfaces we can adapt the degrees along the boundary of that surface. More efficient is a self-cobordism of G with itself with a host graph discretizing the product of G with an interval. It follows from the fact that Euler curvature is zero everywhere for three dimensional geometric graphs, that the odd degree edge set O is a cycle and so a boundary if H is simply connected. A reduction to minimal colouring would imply the four colour theorem. The method is expected to give a reason "why 4 colours suffice" and suggests that every two dimensional geometric graph of arbitrary degree and orientation can be coloured by 5 colours: since the projective plane can not be a boundary of a 3-dimensional graph and because for higher genus surfaces, the interior H is not simply connected, we need in general to embed a surface into a 4-dimensional simply connected graph in order to colour it. This explains the appearance of the chromatic number 5 for higher degree or non-orientable situations, a number we believe to be the upper limit. For every surface type, we construct examples with chromatic number 3,4 or 5, where the construction of surfaces with chromatic number 5 is based on a method of Fisk. We have implemented and illustrated all the topological aspects described in this paper on a computer. So far we still need human guidance or simulated annealing to do the refinements in the higher dimensional host graph.Comment: 81 pages, 48 figure

Equitable Colorings of Corona Multiproducts of Graphs

A graph is equitably $k$-colorable if its vertices can be partitioned into $k$ independent sets in such a way that the number of vertices in any two sets differ by at most one. The smallest $k$ for which such a coloring exists is known as the equitable chromatic number of $G$ and denoted $\chi_{=}(G)$. It is known that this problem is NP-hard in general case and remains so for corona graphs. In "Equitable colorings of Cartesian products of graphs" (2012) Lin and Chang studied equitable coloring of Cartesian products of graphs. In this paper we consider the same model of coloring in the case of corona products of graphs. In particular, we obtain some results regarding the equitable chromatic number for $l$-corona product $G \circ ^l H$, where $G$ is an equitably 3- or 4-colorable graph and $H$ is an $r$-partite graph, a path, a cycle or a complete graph. Our proofs are constructive in that they lead to polynomial algorithms for equitable coloring of such graph products provided that there is given an equitable coloring of $G$. Moreover, we confirm Equitable Coloring Conjecture for corona products of such graphs. This paper extends our results from \cite{hf}.Comment: 14 pages, 1 figur

An algebraic reduction of Hedetniemi's conjecture

For a graph $G$, let $\chi (G)$ denote the chromatic number. In graph theory, the following famous conjecture posed by Hedetniemi has been studied: For two graphs $G$ and $H$, $\chi (G\times H)=\min\{\chi (G),\chi (H)\}$, where $G \times H$ is the tensor product of $G$ and $H$. In this paper, we give a reduction of Hedetniemi's conjecture to an inclusion relation problem on ideals of polynomial rings, and we demonstrate computational experiments for partial solutions of Hedetniemi's conjecture along such a strategy using Gr\"{o}bner basis.Comment: 19 pages, 4 figure

Prime Power and Prime Product Distance Graphs

A graph $G$ is a $k$-prime product distance graph if its vertices can be labeled with distinct integers such that for any two adjacent vertices, the difference of their labels is the product of at most $k$ primes. A graph has prime product number $ppn(G)=k$ if it is a $k$-prime product graph but not a $(k-1)$-prime product graph. Similarly, $G$ is a prime $k$th-power graph (respectively, strict prime $k$th-power graph) if its vertices can be labeled with distinct integers such that for any two adjacent vertices, the difference of their labels is the $j$th power of a prime, for $j \leq k$ (respectively, the $k$th power of a prime exactly). We prove that $ppn(K_n) = \lceil \log_2(n)\rceil - 1$, and for a nonempty $k$-chromatic graph $G$, $ppn(G) = \lceil \log_2(k)\rceil - 1$ or $ppn(G) = \lceil \log_2(k)\rceil$. We determine $ppn(G)$ for all complete bipartite, 3-partite, and 4-partite graphs. We prove that $K_n$ is a prime $k$th-power graph if and only if $n < 7$, and we determine conditions on cycles and outerplanar graphs $G$ for which $G$ is a strict prime $k$th-power graph. We find connections between prime product and prime power distance graphs and the Twin Prime Conjecture, the Green-Tao Theorem, and Fermat's Last Theorem

Nested colourings of graphs

A proper vertex colouring of a graph is \emph{nested} if the vertices of each of its colour classes can be ordered by inclusion of their open neighbourhoods. Through a relation to partially ordered sets, we show that the nested chromatic number can be computed in polynomial time. Clearly, the nested chromatic number is an upper bound for the chromatic number of a graph. We develop multiple distinct bounds on the nested chromatic number using common properties of graphs. We also determine the behaviour of the nested chromatic number under several graph operations, including the direct, Cartesian, strong, and lexicographic product. Moreover, we classify precisely the possible nested chromatic numbers of graphs on a fixed number of vertices with a fixed chromatic number.Comment: 23 pages; 7 figure

Hedetniemi's Conjecture Via Altermatic Number

A $50$ years unsolved conjecture by Hedetniemi [{\it Homomorphisms of graphs and automata, \newblock {\em Thesis (Ph.D.)--University of Michigan}, 1966}] asserts that the chromatic number of the categorical product of two graphs $G$ and $H$ is $\min\{\chi(G),\chi(H)\}$. The present authors [{\it On the chromatic number of general {K}neser hypergraphs. \newblock {\em Journal of Combinatorial Theory, Series B}, 2015.}] introduced the altermatic and the strong altermatic number of graphs as two tight lower bounds for the chromatic number of graphs. In this work, we prove a relaxation of Hedetniemi's conjecture in terms of strong altermatic number. Also, we present a tight lower bound for the chromatic number of the categorical product of two graphs in term of their altermatic and strong altermatic numbers. These results enrich the family of pair graphs $\{G,H\}$ satisfying Hedetniemi's conjecture

The Thue choice number versus the Thue chromatic number of graphs

We say that a vertex colouring $\varphi$ of a graph $G$ is nonrepetitive if there is no positive integer $n$ and a path on $2n$ vertices $v_{1}\ldots v_{2n}$ in $G$ such that the associated sequence of colours $\varphi(v_{1})\ldots\varphi(v_{2n})$ satisfy $\varphi(v_{i})=\varphi(v_{i+n})$ for all $i=1,2,\dots,n$. The minimum number of colours in a nonrepetitive vertex colouring of $G$ is the Thue chromatic number $\pi (G)$. For the case of vertex list colourings the Thue choice number $\pi_{l}(G)$ of $G$ denotes the smallest integer $k$ such that for every list assignment $L:V(G)\rightarrow 2^{\mathbb{N}}$ with minimum list length at least $k$, there is a nonrepetitive vertex colouring of $G$ from the assigned lists. Recently it was proved that the Thue chromatic number and the Thue choice number of the same graph may have an arbitrary large difference in some classes of graphs. Here we give an overview of the known results where we compare these two parameters for several families of graphs and we also give a list of open problems on this topic