scholarly journals On induced subgraphs of the Hamming graph

2020 ◽  
Vol 96 (1) ◽  
pp. 160-166
Author(s):  
Dingding Dong
Keyword(s):  
Induced Subgraphs ◽  
Hamming Graph

Excluding induced subgraphs: Critical graphs

2010 ◽  
Vol 38 (1-2) ◽  
pp. 100-120 ◽  
Author(s):  
József Balogh ◽  
Jane Butterfield
Keyword(s):  
Induced Subgraphs ◽  
Critical Graphs

scholarly journals Degree and neighborhood intersection conditions restricted to induced subgraphs ensuring Hamiltonicity of graphs

2014 ◽  
Vol 06 (03) ◽  
pp. 1450043
Author(s):  
Bo Ning ◽  
Shenggui Zhang ◽  
Bing Chen
Keyword(s):  
Hamilton Cycles ◽  
Induced Subgraphs ◽  
Degree Sum ◽  
Degree Condition

Let claw be the graph K1,3. A graph G on n ≥ 3 vertices is called o-heavy if each induced claw of G has a pair of end-vertices with degree sum at least n, and called 1-heavy if at least one end-vertex of each induced claw of G has degree at least n/2. In this note, we show that every 2-connected o-heavy or 3-connected 1-heavy graph is Hamiltonian if we restrict Fan-type degree condition or neighborhood intersection condition to certain pairs of vertices in some small induced subgraphs of the graph. Our results improve or extend previous results of Broersma et al., Chen et al., Fan, Goodman and Hedetniemi, Gould and Jacobson, and Shi on the existence of Hamilton cycles in graphs.

scholarly journals Induced subgraphs of graphs with large chromatic number. XI. Orientations

2019 ◽  
Vol 76 ◽  
pp. 53-61 ◽  
Author(s):  
Maria Chudnovsky ◽  
Alex Scott ◽  
Paul Seymour
Keyword(s):  
Chromatic Number ◽  
Induced Subgraphs ◽  
Large Chromatic Number

scholarly journals Large induced subgraphs with equated maximum degree

2010 ◽  
Vol 310 (4) ◽  
pp. 742-747 ◽  
Author(s):  
Y. Caro ◽  
R. Yuster
Keyword(s):  
Maximum Degree ◽  
Induced Subgraphs

scholarly journals Edge-Disjoint Induced Subgraphs with Given Minimum Degree

10.37236/2882 ◽  
2013 ◽  
Vol 20 (1) ◽  
Author(s):  
Raphael Yuster
Keyword(s):  
Positive Integer ◽  
Minimum Degree ◽  
Independent Set ◽  
Induced Subgraphs ◽  
Asymptotically Optimal ◽  
Edge Disjoint

Let $h$ be a given positive integer. For a graph with $n$ vertices and $m$ edges, what is the maximum number of pairwise edge-disjoint {\em induced} subgraphs, each having  minimum degree at least $h$? There are examples for which this number is $O(m^2/n^2)$. We prove that this bound is achievable for all graphs with polynomially many edges. For all $\epsilon > 0$, if $m \ge n^{1+\epsilon}$, then there are always $\Omega(m^2/n^2)$ pairwise edge-disjoint induced subgraphs, each having  minimum degree at least $h$. Furthermore, any two subgraphs intersect in an independent set of size at most $1+ O(n^3/m^2)$, which is shown to be asymptotically optimal.

scholarly journals Matrix Partitions with Finitely Many Obstructions

10.37236/976 ◽  
2007 ◽  
Vol 14 (1) ◽  
Author(s):  
Tomás Feder ◽  
Pavol Hell ◽  
Wing Xie
Keyword(s):  
Symmetric Matrix ◽  
Forbidden Subgraph ◽  
Input Graph ◽  
Induced Subgraphs ◽  
Induced Subgraph ◽  
Forbidden Induced Subgraphs ◽  
Extra Effort ◽  
Matrix Partition ◽  
Finite Set ◽  
Forbidden Induced Subgraph

Each $m$ by $m$ symmetric matrix $M$ over $0, 1, *$, defines a partition problem, in which an input graph $G$ is to be partitioned into $m$ parts with adjacencies governed by $M$, in the sense that two distinct vertices in (possibly equal) parts $i$ and $j$ are adjacent if $M(i,j)=1$, and nonadjacent if $M(i,j)=0$. (The entry $*$ implies no restriction.) We ask which matrix partition problems admit a characterization by a finite set of forbidden induced subgraphs. We prove that matrices containing a certain two by two diagonal submatrix $S$ never have such characterizations. We then develop a recursive technique that allows us (with some extra effort) to verify that matrices without $S$ of size five or less always have a finite forbidden induced subgraph characterization. However, we exhibit a six by six matrix without $S$ which cannot be characterized by finitely many induced subgraphs. We also explore the connection between finite forbidden subgraph characterizations and related questions on the descriptive and computational complexity of matrix partition problems.

scholarly journals Norton Algebras of the Hamming Graphs via Linear Characters

2021 ◽  
Vol 28 (2) ◽  
Author(s):  
Jia Huang
Keyword(s):  
Group Theory ◽  
Automorphism Group ◽  
Nonassociative Algebra ◽  
Bilinear Forms ◽  
Distance Regular Graph ◽  
Finite Abelian Groups ◽  
Hamming Graph ◽  
Large Subgroup ◽  
Special Cases ◽  
Hamming Graphs

The Norton product is defined on each eigenspace of a distance regular graph by the orthogonal projection of the entry-wise product. The resulting algebra, known as the Norton algebra, is a commutative nonassociative algebra that is useful in group theory due to its interesting automorphism group. We provide a formula for the Norton product on each eigenspace of a Hamming graph using linear characters. We construct a large subgroup of automorphisms of the Norton algebra of a Hamming graph and completely describe the automorphism group in some cases. We also show that the Norton product on each eigenspace of a Hamming graph is as nonassociative as possible, except for some special cases in which it is either associative or equally as nonassociative as the so-called double minus operation previously studied by the author, Mickey, and Xu. Our results restrict to the hypercubes and extend to the halved and/or folded cubes, the bilinear forms graphs, and more generally, all Cayley graphs of finite abelian groups.

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