Two theorems on the minimum number of intersections for complete graphs

We show that for every sufficiently largen, the number of monotone subsequences of length four in a permutation onnpoints is at least\begin{equation*} \binom{\lfloor{n/3}\rfloor}{4} + \binom{\lfloor{(n+1)/3}\rfloor}{4} + \binom{\lfloor{(n+2)/3}\rfloor}{4}. \end{equation*}Furthermore, we characterize all permutations on [n] that attain this lower bound. The proof uses the flag algebra framework together with some additional stability arguments. This problem is equivalent to some specific type of edge colourings of complete graphs with two colours, where the number of monochromaticK4is minimized. We show that all the extremal colourings must contain monochromaticK4only in one of the two colours. This translates back to permutations, where all the monotone subsequences of length four are all either increasing, or decreasing only.

Download Full-text

Minimal Decompositions of Complete Graphs into Subgraphs with Embeddability Properties

Canadian Journal of Mathematics ◽

10.4153/cjm-1969-109-4 ◽

1969 ◽

Vol 21 ◽

pp. 992-1000 ◽

Cited By ~ 12

Author(s):

L. W. Beineke

Keyword(s):

Projective Plane ◽

Complete Graph ◽

Planar Graphs ◽

Complete Graphs ◽

Minimum Number ◽

Double Torus

Although the problem of finding the minimum number of planar graphs into which the complete graph can be decomposed remains partially unsolved, the corresponding problem can be solved for certain other surfaces. For three, the torus, the double-torus, and the projective plane, a single proof will be given to provide the solutions. The same questions will also be answered for bicomplete graphs.

Download Full-text

Strong chromatic index of products of graphs

Discrete Mathematics & Theoretical Computer Science ◽

10.46298/dmtcs.414 ◽

2007 ◽

Vol Vol. 9 no. 1 (Graph and Algorithms) ◽

Author(s):

Olivier Togni

Keyword(s):

Cartesian Product ◽

Complete Graphs ◽

Chromatic Index ◽

Induced Matching ◽

Colour Class ◽

Strong Chromatic Index ◽

Minimum Number ◽

International Audience

Graphs and Algorithms International audience The strong chromatic index of a graph is the minimum number of colours needed to colour the edges in such a way that each colour class is an induced matching. In this paper, we present bounds for strong chromatic index of three different products of graphs in term of the strong chromatic index of each factor. For the cartesian product of paths, cycles or complete graphs, we derive sharper results. In particular, strong chromatic indices of d-dimensional grids and of some toroidal grids are given along with approximate results on the strong chromatic index of generalized hypercubes.

Download Full-text

Distinct Distances in Graph Drawings

The Electronic Journal of Combinatorics ◽

10.37236/831 ◽

2008 ◽

Vol 15 (1) ◽

Cited By ~ 5

Author(s):

Paz Carmi ◽

Vida Dujmović ◽

Pat Morin ◽

David R. Wood

Keyword(s):

Maximum Degree ◽

Combinatorial Geometry ◽

Complete Graphs ◽

Bounded Degree ◽

Cartesian Products ◽

Bounded Treewidth ◽

Line Drawings ◽

Straight Line ◽

Minimum Number ◽

Complete Bipartite Graphs

The distance-number of a graph $G$ is the minimum number of distinct edge-lengths over all straight-line drawings of $G$ in the plane. This definition generalises many well-known concepts in combinatorial geometry. We consider the distance-number of trees, graphs with no $K^-_4$-minor, complete bipartite graphs, complete graphs, and cartesian products. Our main results concern the distance-number of graphs with bounded degree. We prove that $n$-vertex graphs with bounded maximum degree and bounded treewidth have distance-number in ${\cal O}(\log n)$. To conclude such a logarithmic upper bound, both the degree and the treewidth need to be bounded. In particular, we construct graphs with treewidth $2$ and polynomial distance-number. Similarly, we prove that there exist graphs with maximum degree $5$ and arbitrarily large distance-number. Moreover, as $\Delta$ increases the existential lower bound on the distance-number of $\Delta$-regular graphs tends to $\Omega(n^{0.864138})$.

Download Full-text

ON METRIC DIMENSION OF FUNCTIGRAPHS

Discrete Mathematics Algorithms and Applications ◽

10.1142/s1793830912500607 ◽

2013 ◽

Vol 05 (04) ◽

pp. 1250060 ◽

Cited By ~ 2

Author(s):

LINDA EROH ◽

CONG X. KANG ◽

EUNJEONG YI

Keyword(s):

Connected Graph ◽

Metric Dimension ◽

Complete Graphs ◽

Minimum Number ◽

Vertex Set

The metric dimension of a graph G, denoted by dim (G), is the minimum number of vertices such that each vertex is uniquely determined by its distances to the chosen vertices. Let G1and G2be disjoint copies of a graph G and let f : V(G1) → V(G2) be a function. Then a functigraphC(G, f) = (V, E) has the vertex set V = V(G1) ∪ V(G2) and the edge set E = E(G1) ∪ E(G2) ∪ {uv | v = f(u)}. We study how metric dimension behaves in passing from G to C(G, f) by first showing that 2 ≤ dim (C(G, f)) ≤ 2n - 3, if G is a connected graph of order n ≥ 3 and f is any function. We further investigate the metric dimension of functigraphs on complete graphs and on cycles.

Download Full-text

Hajós and Ore Constructions for Digraphs

The Electronic Journal of Combinatorics ◽

10.37236/8942 ◽

2020 ◽

Vol 27 (1) ◽

Author(s):

Jørgen Bang-Jensen ◽

Thomas Bellitto ◽

Thomas Schweser ◽

Michael Stiebitz

Keyword(s):

Type Theorem ◽

Complete Graphs ◽

Color Class ◽

Minimum Number ◽

Dichromatic Number

The dichromatic number $\overrightarrow{\chi}(D)$ of a digraph $D$ is the minimum number of colors needed to color the vertices of $D$ such that each color class induces an acyclic subdigraph of $D$. A digraph $D$ is $k$-critical if $\overrightarrow{\chi}(D) = k$ but $\overrightarrow{\chi}(D') < k$ for all proper subdigraphs $D'$ of $D$. We examine methods for creating infinite families of critical digraphs, the Dirac join and the directed and bidirected Hajós join. We prove that a digraph $D$ has dichromatic number at least $k$ if and only if it contains a subdigraph that can be obtained from bidirected complete graphs on $k$ vertices by directed Hajós joins and identifying non-adjacent vertices. Building upon that, we show that a digraph $D$ has dichromatic number at least $k$ if and only if it can be constructed from bidirected $K_k$'s by using directed and bidirected Hajós joins and identifying non-adjacent vertices (so called Ore joins), thereby transferring a well-known result of Urquhart to digraphs. Finally, we prove a Gallai-type theorem that characterizes the structure of the low vertex subdigraph of a critical digraph, that is, the subdigraph, which is induced by the vertices that have in-degree $k-1$ and out-degree $k-1$ in $D$.

Download Full-text

The conjugacy class graphs of non-abelian 3-groups

Malaysian Journal of Fundamental and Applied Sciences ◽

10.11113/mjfas.v16n3.1453 ◽

2020 ◽

Vol 16 (3) ◽

pp. 297-299

Author(s):

Athirah Zulkarnain ◽

Nor Haniza Sarmin ◽

Hazzirah Izzati Mat Hassim

Keyword(s):

Conjugacy Class ◽

Chromatic Number ◽

Simple Graph ◽

Conjugacy Classes ◽

Prime Power Order ◽

Complete Graphs ◽

Dominating Number ◽

Minimum Number ◽

Vertex Set ◽

Class Graph

A graph is formed by a pair of vertices and edges. It can be related to groups by using the groups’ properties for its vertices and edges. The set of vertices of the graph comprises the elements or sets from the group while the set of edges of the graph is the properties and condition for the graph. A conjugacy class of an element is the set of elements that are conjugated with . Any element of a group , labelled as , is conjugated to if it satisfies for some elements in with its inverse . A conjugacy class graph of a group is defined when its vertex set is the set of non-central conjugacy classes of . Two distinct vertices and are connected by an edge if and only if their cardinalities are not co-prime, which means that the order of the conjugacy classes of and have common factors. Meanwhile, a simple graph is the graph that contains no loop and no multiple edges. A complete graph is a simple graph in which every pair of distinct vertices is adjacent. Moreover, a -group is the group with prime power order. In this paper, the conjugacy class graphs for some non-abelian 3-groups are determined by using the group’s presentations and the definition of conjugacy class graph. There are two classifications of the non-abelian 3-groups which are used in this research. In addition, some properties of the conjugacy class graph such as the chromatic number, the dominating number, and the diameter are computed. A chromatic number is the minimum number of vertices that have the same colours where the adjacent vertices have distinct colours. Besides, a dominating number is the minimum number of vertices that is required to connect all the vertices while a diameter is the longest path between any two vertices. As a result of this research, the conjugacy class graphs of these groups are found to be complete graphs with chromatic number, dominating number and diameter that are equal to eight, one and one, respectively.

Download Full-text

Cops, robbers, and barricades

10.32920/ryerson.14664981.v1 ◽

2021 ◽

Author(s):

Erin Kathleen McKenna Meger

Keyword(s):

Complete Graphs ◽

New Variant ◽

Cops And Robbers ◽

Minimum Number ◽

Compare And Contrast

Cops, Robbers, and Barricades is a new variant of the game on graphs, Cops and Robbers. In this variant, the robber may build barricades that restrict the movements of the cops. The minimum number of cops required to capture the robber on a graph G is called the barricade-cop number, denoted cB(G). If cB(G) = 1, then G is called barricade-cop-win. The game can be generalized so that the robber may build b(k)-many barricades on vertices during her kth turn, in accordance with barricade rules that dictate the permissible positions of these barricades. The barricade-cop number is determined exactly for complete graphs, cycles, and paths, and we provide bounds on trees and locally-path-like graphs. We compare and contrast variants on the barricade rules, and give an algorithmic characterization of barricade-cop-win graphs with any set of barricade rules.

Download Full-text

Cops, robbers, and barricades

10.32920/ryerson.14664981 ◽

2021 ◽

Author(s):

Erin Kathleen McKenna Meger

Keyword(s):

Complete Graphs ◽

New Variant ◽

Cops And Robbers ◽

Minimum Number ◽

Compare And Contrast

Cops, Robbers, and Barricades is a new variant of the game on graphs, Cops and Robbers. In this variant, the robber may build barricades that restrict the movements of the cops. The minimum number of cops required to capture the robber on a graph G is called the barricade-cop number, denoted cB(G). If cB(G) = 1, then G is called barricade-cop-win. The game can be generalized so that the robber may build b(k)-many barricades on vertices during her kth turn, in accordance with barricade rules that dictate the permissible positions of these barricades. The barricade-cop number is determined exactly for complete graphs, cycles, and paths, and we provide bounds on trees and locally-path-like graphs. We compare and contrast variants on the barricade rules, and give an algorithmic characterization of barricade-cop-win graphs with any set of barricade rules.

Download Full-text

On injective chromatic polynomials of graphs

Discrete Mathematics Algorithms and Applications ◽

10.1142/s1793830915500354 ◽

2015 ◽

Vol 07 (03) ◽

pp. 1550035

Author(s):

Anjaly Kishore ◽

M. S. Sunitha

Keyword(s):

Chromatic Number ◽

Discrete Math ◽

Chromatic Polynomial ◽

Complete Graphs ◽

Common Neighbor ◽

Chromatic Polynomials ◽

Wheel Graphs ◽

Minimum Number

The injective chromatic number χi(G) [G. Hahn, J. Kratochvil, J. Siran and D. Sotteau, On the injective chromatic number of graphs, Discrete Math. 256(1–2) (2002) 179–192] of a graph G is the minimum number of colors needed to color the vertices of G such that two vertices with a common neighbor are assigned distinct colors. The nature of the coefficients of injective chromatic polynomials of complete graphs, wheel graphs and cycles is studied. Injective chromatic polynomial on operations like union, join, product and corona of graphs is obtained.

Download Full-text