It was recently proved that every planar graph is a subgraph of the strong product of a path and a graph with bounded treewidth. This paper surveys generalisations of this result for graphs on surfaces, minor-closed classes, various non-minor-closed classes, and graph classes with polynomial growth. We then explore how graph product structure might be applicable to more broadly defined graph classes. In particular, we characterise when a graph class defined by a cartesian or strong product has bounded or polynomial expansion. We then explore graph product structure theorems for various geometrically defined graph classes, and present several open problems.
In this short note we show that every connected $2$-edge coloured cubic graph admits an $10$-colouring. This lowers the best known upper bound for the chromatic number of connected $2$-edge coloured cubic graphs.
In this note, we show that a complete $k$-partite graph is the only graph with clique number $k$ among all degree-equivalent simple graphs. This result gives a lower bound on the clique number, which is sharper than existing bounds on a large family of graphs.
It is well-known that 1-planar graphs have minimum degree at most 7, and not hard to see that some 1-planar graphs have minimum degree exactly 7. In this note we show that any such 1-planar graph has at least 24 vertices, and this is tight.
The symmetric version of Fitch's xenology relation coincides with class of complete multipartite graph and thus cannot convey any non-trivial phylogenetic information.
This note resolves an open problem asked by Bezrukov in the open problem session of IWOCA 2014. It shows an equivalence between regular graphs and graphs for which a sequence of invariants presents some symmetric property. We extend this result to a few other sequences.
Recently Naserasr, Sopena, and Zaslavsky [R. Naserasr, É. Sopena, T. Zaslavsky,Homomorphisms of signed graphs: An update, arXiv: 1909.05982v1 [math.CO] 12 Sep 2019.] published a report on closed walks in signed graphs. They gave a characterization of the sets of closed walks in a graph $G$ which corespond to the set of negative walks in some signed graph on $G$. In this note we show that their characterization is not valid and give a new characterization.
In this note we show every orientation of a connected cubic graph admits an oriented 8-colouring. This lowers the best-known upper bound for the chromatic number of the family of orientations of connected cubic graphs. We further show that every such oriented graph admits a 2-dipath 7-colouring. These results imply that either the chromatic number for the family of oriented connected cubic graphs equals the 2-dipath chromatic number or the long-standing conjecture of Sopena [Journal of Graph Theory 25:191-205 1997] regarding the chromatic number of orientations of connected cubic graphs is fal…
Given two regular graphs with consistent rotation maps, we produce a constructive method for a consistent rotation map on their Cartesian product. This method will be given as a simple set of rules of addition and table look ups. We assume that the combinatorial construction of both consistent rotation maps has occurred before we construct the Cartesian product.
We prove that there is an absolute constant $C>0$ so that for every natural $n$ there exists a triangle-free \emph{regular} graph with no independent set of size at least $C\sqrt{n\log n}$.
In this short note, we show two NP-completeness results regarding the \emph{simultaneous representation problem}, introduced by Lubiw and Jampani. The simultaneous representation problem for a given class of intersection graphs asks if some $k$ graphs can be represented so that every vertex is represented by the same interval in each representation. We prove that it is NP-complete to decide this for the class of interval and circular-arc graphs in the case when $k$ is a part of the input and graphs are not in a sunflower position.
We give a simple proof that every $n$-vertex graph $d$-regular graph that does not contain a fixed bipartite graph as a subgraph has an induced matching of size $Ω((n/d)(\log d))$.
These are some personal notes about the pursuit game of Cops and Robbers that I made starting in 2007. More old and new problems (and some solutions) will be added in future versions of these notes.
A fall coloring of a graph G is a proper coloring of G with k colors such that each vertex sees all k colors on its closed neighborhood. In this short note, we characterize all fall colorings of Kneser graphs of type KG(n,2).
Among all simple 2-connected graphs, and among all $θ$-graphs, the graphs with the minimum algebraic connectivity are completely determined, respectively.
In this short note, the purpose is to provide an upper bound for the b-chromatic number of Kneser graphs. Our bound improves the upper bound that was presented by Balakrishnan and Kavaskar in [b-coloring of Kneser graphs, Discrete Appl. Math. 160 (2012), 9-14].
In this note, we consider connected graphs with exactly two main eigenvalues. We will give several constructions for them, and as a consequence we show a family of those graphs with an unbounded number of distinct valencies.
We consider a set of cliques in any multipartite graph with two vertices in each part. Moreover, we construct a class of peculiar polytopes. Key words: multipartite graph, clique, polytope.
We study the height of a spanning tree $T$ of a graph $G$ obtained by starting with a single vertex of $G$ and repeatedly selecting, uniformly at random, an edge of $G$ with exactly one endpoint in $T$ and adding this edge to $T$.
The family of Directed Acyclic Graphs as well as some related graphs are analyzed with respect to extremal behavior in relation with the family of intersection graphs for families of boxes with transverse intersection.
In this paper, we obtain a sufficient condition for the existence of parity factors in a regular graph in terms of edge-connectivity. Moreover, we also show that our condition is sharp.
In 1994, Cornelis Hoede and Xueliang Li introduced the clique polynomial of a graph. Also, a theorem for the edge subgraph expansion for clique polynomials. In this note we present a counter example for it and explain which case it could be a valid theorem.
A connected $n$-chromatic graph $G$ is double-critical if for all the edges $xy$ of $G$, the graph $G-x-y$ is $(n-2)$-chromatic. In 1966, Erd\H os and Lovász conjectured that the only double-critical $n$-chromatic graph is $K_n$. This conjecture remains unresolved for $n \ge 6.$ In this short note, we verify this conjecture for claw-free graphs $G$ of chromatic number $6$.
A simplified construction is presented for Komjáth's result that for every uncountable cardinal $κ$, there are $2^κ$ graphs of size $κ$ none of them being a minor of another.
Even-hole-free graphs are a graph class of much interest. Foley et al. [Graphs Comb. 36(1): 125-138 (2020)] have recently studied $(4K_1, C_4, C_6)$-free graphs, which form a subclass of even-hole-free graphs. Specifically, Foley et al. have shown an algorithm for coloring these graphs via bounded clique-width if they contain a $C_7$. In this note, we give a simpler and much faster algorithm via a more restrictive graph parameter, neighborhood diversity.
This note answers a question posed by Levine in arkiv:math.CO/0906.2809. The main result is Theorem 1 which shows that under certain circumstances a critical group of a directed graph is the quotient of a critical group of its directed line graph.
Let $r$ be an odd integer, and $k$ an even integer. In this note, we present $r$-regular graphs which have no $\{k,r-k\}$-factors for all $1\le k\le {r\over2}-1$. This gives a negative answer to a problem posed by Akbari and Kano recently.
Neumann-Lara and Škrekovski conjectured that every planar digraph is $2$-colourable. We show that this conjecture is equivalent to the more general statement that all oriented $K_5$-minor-free graphs are $2$-colourable.
A proof that every outerplanar graph is Δ+2 colorable. This is slightly stronger then an unpublished result of Wang Shudong, Ma Fangfang, Xu Jin, and Yan Lijun proving the same for 2-connected outerplanar graphs.
We give some properties of skew spectrum of a graph, especially, we answer negatively a problem concerning the skew characteristic polynomial and matching polynomial in [M. Cavers et al., Skew-adjacency matrices of graphs, Linear Algebra Appl. 436 (2012) 4512--4529].
We show that any n-vertex graph without even cycles of length at most 2k has at most 1/2(n^{1 + 1/k}) + O(n) edges, and polarity graphs of generalized polygons show that this is asymptotically tight when k = 2,3,5.
The $t$-e.c. and pseudo-random property are typical properties of random graphs. In this note, we study the gap between them which has not been studied well. As a main result, we give the first explicit construction of infinite families of t-e.c. graphs which are not families of best possible pseudo-random graphs.
The bondage number b(G) of a graph G is the smallest number of edges whose removal from G results in a graph with larger domination number. In this paper we present new upper bounds for b(G) in terms of girth, order and Euler characteristic.
A \emph{$k$--bisection} of a bridgeless cubic graph $G$ is a $2$--colouring of its vertex set such that the colour classes have the same cardinality and all connected components in the two subgraphs induced by the colour classes have order at most $k$. Ban and Linial conjectured that {\em every bridgeless cubic graph admits a $2$--bisection except for the Petersen graph}. In this note, we prove Ban--Linial's conjecture for claw--free cubic graphs.
A rainbow matching in an edge-coloured graph is a matching such that its edges have distinct colours. We show that every properly edge-coloured graph $G$ with $|G| \ge (9δ(G) -5)/2$ has a rainbow matching of size $δ(G)$, improving a result of Diemunsch et al.
This is a note on three graph parameters motivated by the Euler-Poincare characteristic for simplicial complex. We show those three graph parameters of a given connected graph $G$ is greater than or equal to that of the complete graph with $\max(h(G),χ(G))$ vertices. This will yield three different simultaneous upperbounds of both the hadwiger number and chromatic number by means of the number of particular types of induced subgraphs. Some applications to Hadwiger's Conjecture is also discussed.
In this paper, we provide few results on the group distance magic labeling of lexicographic product and direct product of two graphs. We also prove some necessary conditions for a graph to be group distance magic and provide a characterization for a tree to be group distance magic.
For a simple graph $G$ and a real number $α$ $\left(α\neq 0,1\right) $ the graph invariant $s_α\left(G\right) $ is equal to the sum of powers of signless Laplacian eigenvalues of $G$. In this note, we present some new bounds on $s_α\left(G\right) $. As a result of these bounds, we also give some results on incidence energy.
We present a family of 3-connected cubic planar Hamiltonian graphs with an exponential number of steps required by Thomason's algorithm. The base of the exponent is approximately $1.1812...$, which exceeds previous results in the area.
The harmonic index of a graph $G$ is defined as the sum of weights $\frac{2}{deg(v) + deg(u)}$ of all edges $uv$ of $E (G)$, where $deg (v)$ denotes the degree of a vertex $v$ in $V (G)$. In this note we generalize results of [L. Zhong, The harmonic index on graphs, Appl. Math. Lett. 25 (2012), 561--566] and establish some upper and lower bounds on the harmonic index of $G$.
We show that the independence complex of a chordal graph is contractible if and only if this complex is dismantlable (strong collapsible) and it is homotopy equivalent to a sphere if and only if its core is a cross-polytopal sphere. The proof uses the properties of tree models of chordal graphs.
We prove a conjecture by Stefan Kohl on the existence of triples of permutations of bounded degree with prescribed orders and product 1. This result leads to an existence result for covers of the complex projective line with bounded degree and prescribed ramification indices.
In a recent paper, Grau et al. (2017) studied the zero divisor graphs of the ring of Lipschitz integers modulo $n$, and computed the domination number of the undirected zero divisor graph of the ring of Lipschitz integers modulo $n$. But the case $n$ is a power of prime numbers remained open. In this note, this problem is solved. We also show the automorphism group of the zero divisor graph of the ring of Lipschitz integers modulo $2^s$.
For a graph $G$, let $τ(G)$ be the maximum number of colors such that there exists an edge-coloring of $G$ with no two color classes being isomorphic. We investigate the behavior of $τ(G)$ when $G=G(n, p)$ is the classical Erdős-Rényi random graph.
The concept of the \emph{Gutman index}, denoted $Gut(G)$ was introduced for a connected undirected graph $G$. In this note we apply the concept to the underlying graphs of the family of Jaco graphs, (\emph{directed graphs by definition}), and describe a recursive formula for the \emph{Gutman index} $Gut(J^*_{n+1}(x)).$ We also determine the \emph{Gutman index} for the trivial \emph{edge-joint} between Jaco graphs.
Let $G$ be a graph of radius $r$ and diameter $d$ with $d\leq 2r-2$. We give a new proof that $G$ contains a cycle of length at least $4r-2d$, i.e. for its circumference it holds $c(G)\geq 4r-2d$.
This paper is devoted to present two counterexamples to the theorem from \cite{MK} Maria R., Katherine T. M., Bernardo S. M., Extremal graphs with bounded vertex bipartiteness number, Linear Algebra Appl. 493 (2016) 28-36. Moreover, the corrected theorem and proof are presented.
