An odd characterization of the generalized odd graphs

We show that any connected regular graph with d+1 distinct eigenvalues and odd-girth 2d+1 is distance-regular, and in particular that it is a generalized odd graph.     

Characterization of the odd graphs Ok by parameters

In this note, we settle a problem of N. Biggs [4, p.80] by showing that for each k, no distance regular graph non-isomorphic to the odd graph O_k can have the same parameters as O_k. A related characterization of certain graphs associated with the Johnson scheme J(2k+1, k) is also given.     

Homomorphisms of graphs into odd cycles

We give a class of graphs G for which there exists a homomorphism (= adjacency preserving map) from V(G) to V(C), where C is the shortest odd cycle in G, thereby extending a result of Albertson, Catlin, and Gibbons. Our class of graphs is characterized by the following property: For each odd subdivision G′ of G there exists a homomorphic map from V(G′) to V(C), where C′ is the shortest odd cycle of G′.     

Regular Steinhaus graphs of odd degree

A Steinhaus matrix is a binary square matrix of size n which is symmetric, with a diagonal of zeros, and whose upper-triangular coefficients satisfy a_i,j=a_i−1,j−1+a_i−1,j for all 2?i<j?n. Steinhaus matrices are determined by their first row. A Steinhaus graph is a simple graph whose adjacency matrix is a Steinhaus matrix. We give a short new proof of a theorem, due to Dymacek, which states that even Steinhaus graphs, i.e. those with all vertex degrees even, have doubly-symmetric Steinhaus matrices. In 1979 Dymacek conjectured that the complete graph on two vertices K₂ is the only regular Steinhaus graph of odd degree. Using Dymacek’s theorem, we prove that if (a_i,j)_1?i,j?n is a Steinhaus matrix associated with a regular Steinhaus graph of odd degree then its sub-matrix (a_i,j)_2?i,j?n−1 is a multi-symmetric matrix, that is a doubly-symmetric matrix where each row of its upper-triangular part is a symmetric sequence. We prove that the multi-symmetric Steinhaus matrices of size n whose Steinhaus graphs are regular modulo 4, i.e. where all vertex degrees are equal modulo 4, only depend on parameters for all even numbers n, and on parameters in the odd case. This result permits us to verify Dymacek’s conjecture up to 1500 vertices in the odd case.     

图(C)2n的奇优美及其奇强协调性

该文定义了图(C)2n,并研究了该图的奇优美和奇强协调性.利用构造法分别给出了图(C)2n在n=4k(k≥2)、n=4k+2时的奇优美算法,在n=4kk≥2)时,的奇强协调算法,进而证明了图(C)2n在n=2k(k≥3)时是奇优美图,在n=4k(k≥2)时是奇强协调图等结论,从而推动了对图的奇优美性和奇强协调性的研究.最后提出猜想:当n=4k+2时,图(C)2n不是奇强协调图.     

Regular odd rings and non-planar graphs

In a previous paper we have announced that a graph is non-planar if and only if it contains a maximal, strict, compact, odd ring. Little has conjectured that the compactness condition may be removed. Chernyak has now published a proof of this conjecture. However, it is difficult to test a ring for maximality. In this paper we show that for odd rings of size five or greater, the condition of maximality may be replaced by a new one called regularity. Regularity is an easier condition to diagnose than is maximality.     

Odd connection and odd vertex-connection of graphs

In the last years, connection concepts such as rainbow connection and proper connection appeared in graph theory and received a lot of attention. In this paper, we present a general concept of connection in graphs. As a particular case, we introduce the odd connection number and the odd vertex-connection number of a graph. Furthermore, we compute and study the odd connection number and the odd vertex-connection number of graphs of various graph classes.     

Some results on odd factors of graphs

A {1, 3, …,2n ? 1}-factor of a graph G is defined to be a spanning subgraph of G, each degree of whose vertices is one of {1, 3, …, 2n ? 1}, where n is a positive integer. In this paper, we give a sufficient condition for a graph to have a {1, 3, …, 2n ? 1}-factor.     

Regular path decompositions of odd regular graphs

Kotzig asked in 1979 what are necessary and sufficient conditions for a d‐regular simple graph to admit a decomposition into paths of length d for odd d>3. For cubic graphs, the existence of a 1‐factor is both necessary and sufficient. Even more, each 1‐factor is extendable to a decomposition of the graph into paths of length 3 where the middle edges of the paths coincide with the 1‐factor. We conjecture that existence of a 1‐factor is indeed a sufficient condition for Kotzig's problem. For general odd regular graphs, most 1‐factors appear to be extendable and we show that for the family of simple 5‐regular graphs with no cycles of length 4, all 1‐factors are extendable. However, for d>3 we found infinite families of d‐regular simple graphs with non‐extendable 1‐factors. Few authors have studied the decompositions of general regular graphs. We present examples and open problems; in particular, we conjecture that in planar 5‐regular graphs all 1‐factors are extendable. © 2009 Wiley Periodicals, Inc. J Graph Theory 63: 114–128, 2010     

Subconstituents of orthogonal graphs of odd characteristic

In this paper, the subconstituents of the isotropic orthogonal graphs over finite fields of odd characteristic are studied. The first subconstituent is strongly regular, while the second subconstituent is edge-regular. The full automorphism groups of these two subconstituents have also been determined.     

Packing graphs with odd and even trees

The Gyárfás-Lehel tree-packing conjecture asserts that any sequence T₁, T₂, …, T_n^?1 of trees with 1, 2, …, n - 1 edges packs into the complete graph K_n on n vertices. The present paper examines two conjectures that jointly imply the Gyárfás-Lehel conjecture: 1. For n even, any T₁, T₃, …, T_n^?1 pack into the half-complete graph Hn on n vertices.2. For n odd, any T₂, T₄, …, T_n^?1 pack into the half-complete graph Hn on n vertices. The H_n are uniquely defined by their degree sequences: H_n and H_n+1 are complements in K_n+1. It is shown that H_n and T_n+1 pack into H_n+2 if T_n+1 is a double star, unimodal triple star, interior-3 caterpillar, or scorpion. Hence Conjectures 1 and 2 are true for these specialized types of trees. The conjectures are also valid for all trees when n ≤ 9, so that the Gyárfás-Lehel conjecture holds for n ≤ 9.     

On the odd harmonious graphs with applications

The necessary conditions for the existence of odd harmonious labelling of graph are obtained. A cycle C _n is odd harmonious if and only if n≡0 (mod 4). A complete graph K _n is odd harmonious if and only if n=2. A complete k-partite graph K(n ₁,n ₂,…,n _k ) is odd harmonious if and only if k=2. A windmill graph K _n ^t is odd harmonious if and only if n=2. The construction ways of odd harmonious graph are given. We prove that the graph ∨ _i=1 ⁿ G _i , the graph G(+r ₁,+r ₂,…,+r _p ), the graph $\bar{K_{m}}+_{0}P_{n}+_{e}\bar{K_{t}}$ , the graph G∪(X+∪ _k=1 ⁿ Y _k ), some trees and the product graph P _m ×P _n etc. are odd harmonious. The odd harmoniousness of graph can be used to solve undetermined equation.     

Smallest regular graphs with prescribed odd girth

The odd girth of a graph G gives the length of a shortest odd cycle in G. Let f(k,g) denote the smallest n such that there exists a k-regular graph of order n and odd girth g. The exact values of f(k,g) are determined if one of the following holds:

• (i) k > 2g ?5 and k is a prime number,
• (ii) k > (2?(g + 1)/4? ?1)², and
• (iiii) k is a perfect square.

     

A Family of Non-quasiprimitive Graphs Constructed From PSU3（q） Where q Is Odd

Let Г be a simple connected graph and let G be a group of automorphisms of Г. Г is said to be （G, 2）-arc transitive if G is transitive on the 2-arcs of Г. It has been shown that there exists a family of non-quasiprimitive （PSU3（q）, 2）-arc transitive graphs where q = 2^3m with m an odd integer. In this paper we investigate the case where q is an odd prime power.     

Vertex colorings of graphs without short odd cycles

Motivated by the work of Ne?et?il and Rödl on “Partitions of vertices” we are interested in obtaining some quantitative extensions of their result. In particular, given a natural number r and a graph G of order m with odd girth g, we show the existence of a graph H with odd girth at least g and order that is polynomial in m such that every r‐coloring of the vertices of H yields a monochromatic and induced copy of G. © 2010 Wiley Periodicals, Inc. J Graph Theory 68: 255‐264, 2011     

Matchings in graphs of odd regularity and girth

We establish lower bounds on the matching number of graphs of given odd regularity d$d$ and odd girth g$g$, which are sharp for many values of d$d$ and g$g$. For d=g=5$d=g=5$, we characterize all extremal graphs.