Graphs Of Girth Research Articles

The Distance-t Chromatic Index of Graphs

We consider two graph colouring problems in which edges at distance at most t are given distinct colours, for some fixed positive integer t. We obtain two upper bounds for the distance-t chromatic index, the least number of colours necessary for such a colouring. One is a bound of (2-ε)Δt for graphs of maximum degree at most Δ, where ε is some absolute positive constant independent of t. The other is a bound of O(Δt/log Δ) (as Δ → ∞) for graphs of maximum degree at most Δ and girth at least 2t+1. The first bound is an analogue of Molloy and Reed's bound on the strong chromatic index. The second bound is tight up to a constant multiplicative factor, as certified by a class of graphs of girth at least g, for every fixed g ≥ 3, of arbitrarily large maximum degree Δ, with distance-t chromatic index at least Ω(Δt/log Δ).

Lower bounds for local approximation

In the study of deterministic distributed algorithms, it is commonly assumed that each node has a unique O (log n )-bit identifier. We prove that for a general class of graph problems, local algorithms (constant-time distributed algorithms) do not need such identifiers: a port numbering and orientation is sufficient. Our result holds for so-called simple PO- checkable graph optimisation problems ; this includes many classical packing and covering problems such as vertex covers, edge covers, matchings, independent sets, dominating sets, and edge dominating sets. We focus on the case of bounded-degree graphs and show that if a local algorithm finds a constant-factor approximation of a simple PO-checkable graph problem with the help of unique identifiers, then the same approximation ratio can be achieved on anonymous networks. As a corollary of our result, we derive a tight lower bound on the local approximability of the minimum edge dominating set problem . By prior work, there is a deterministic local algorithm that achieves the approximation factor of 4--1/⌊Δ/2⌋ in graphs of maximum degree Δ. This approximation ratio is known to be optimal in the port-numbering model—our main theorem implies that it is optimal also in the standard model in which each node has a unique identifier. Our main technical tool is an algebraic construction of homogeneously ordered graphs : We say that a graph is (α, r )-homogeneous if its nodes are linearly ordered so that an α fraction of nodes have pairwise isomorphic radius- r neighbourhoods. We show that there exists a finite (α, r )-homogeneous 2 k -regular graph of girth at least g for any α < 1 and any r , k , and g .

Independent domination in subcubic bipartite graphs of girth at least six

A set S of vertices of a graph G is an independent dominating set of G if S is an independent set and every vertex not in S is adjacent to a vertex in S. The independent domination number of G, denoted by i(G), is the minimum cardinality of an independent dominating set of G. In this paper, we show that if G is a bipartite cubic graph of order n and of girth at least 6, then i(G)≤4n/11.

Linking rings structures and tetravalent semisymmetric graphs

In this paper, we introduce LR structures, a new and interesting form of symmetry in graphs. LR structures are motivated by the search for semisymmetric graphs of degree 4. We show that all semisymmetric graphs of girth and degree 4 can be constructed in a simple way from LR structures. We then show several ways in which LR structures can be constructed or found.

Super cyclically edge-connected vertex-transitive graphs of girth at least 5

A cyclic edge-cut of a graph G is an edge set, the removal of which separates two cycles. If G has a cyclic edge-cut, then it is called cyclically separable. We call a cyclically separable graph super cyclically edge-connected, in short, super-λc, if the removal of any minimum cyclic edge-cut results in a component which is a shortest cycle. In [Zhang, Z., Wang, B.: Super cyclically edge-connected transitive graphs. J. Combin. Optim., 22, 549–562 (2011)], it is proved that a connected vertex-transitive graph is super-λc if G has minimum degree at least 4 and girth at least 6, and the authors also presented a class of nonsuper-λc graphs which have degree 4 and girth 5. In this paper, a characterization of k (k ≥ 4)-regular vertex-transitive nonsuper-λc graphs of girth 5 is given. Using this, we classify all k (k ≥ 4)-regular nonsuper-λc Cayley graphs of girth 5, and construct the first infinite family of nonsuper-λc vertex-transitive non-Cayley graphs.

Additive Coloring of Planar Graphs

An additive coloring of a graph G is an assignment of positive integers \({\{1,2,\ldots ,k\}}\) to the vertices of G such that for every two adjacent vertices the sums of numbers assigned to their neighbors are different. The minimum number k for which there exists an additive coloring of G is denoted by \({\eta (G)}\) . We prove that \({\eta (G) \, \leqslant \, 468}\) for every planar graph G. This improves a previous bound \({\eta (G) \, \leqslant \, 5544}\) due to Norin. The proof uses Combinatorial Nullstellensatz and the coloring number of planar hypergraphs. We also demonstrate that \({\eta (G) \, \leqslant \, 36}\) for 3-colorable planar graphs, and \({\eta (G) \, \leqslant \, 4}\) for every planar graph of girth at least 13. In a group theoretic version of the problem we show that for each \({r \, \geqslant \, 2}\) there is an r-chromatic graph G r with no additive coloring by elements of any abelian group of order r.

Near Packings of Graphs

A packing of a graph $G$ is a set $\{G_1,G_2\}$ such that $G_1\cong G$, $G_2\cong G$, and $G_1$ and $G_2$ are edge disjoint subgraphs of $K_n$. Let $\mathcal{F}$ be a family of graphs. A near packing admitting $\mathcal{F}$ of a graph $G$ is a generalization of a packing. In a near packing admitting $\mathcal{F}$, the two copies of $G$ may overlap so the subgraph defined by the edges common to both copies is a member of $\mathcal{F}$. In the paper we study three families of graphs (1) $\mathcal{E}_k$ -- the family of all graphs with at most $k$ edges, (2) $\mathcal{D}_k$ -- the family of all graphs with maximum degree at most $k$, and (3) $\mathcal{C}_k$ -- the family of all graphs that do not contain a subgraph of connectivity greater than or equal to $k+1$. By $m(n,\mathcal{F})$ we denote the maximum number $m$ such that each graph of order $n$ and size less than or equal to $m$ has a near-packing admitting $\mathcal{F}$. It is well known that $m(n,\mathcal{C}_0)=m(n,\mathcal{D}_0)=m(n,\mathcal{E}_0)=n-2$ because a near packing admitting $\mathcal{C}_0$, $\mathcal{D}_0$ or $\mathcal{E}_0$ is just a packing. We prove some generalization of this result, namely we prove that $ m(n,\mathcal{C}_k)\approx (k+1)n$, $ m(n,\mathcal{D}_1)\approx \frac{3}{2}n$, $ m(n,\mathcal{D}_2)\approx 2n$. We also present bounds on $m(n,\mathcal{E}_k)$. Finally, we prove that each graph of girth at least five has a near packing admitting $\mathcal{C}_1$ (i.e. a near packing admitting the family of the acyclic graphs).

Families of small regular graphs of girth 7

Abstract We present an explicit construction of small ( q + 1 , 7 ) -graphs for q a prime power that is based on some combinatorial properties of the incidence graphs of generalized quadrangles of order ( q , q ) .

Constructions of biregular cages of girth five

Abstract Let 2 ⩽ r m and g be positive integers. An ( { r , m } ; g ) –graph (or biregular graph) is a graph with degree set { r , m } and girth g, and an ( { r , m } ; g ) –cage (or biregular cage) is an ( { r , m } ; g ) –graph of minimum order n ( { r , m } ; g ) . If m = r + 1 , an ( { r , m } ; g ) –cage is said to be a semiregular cage. In this extended abstract we construct two infinite families of biregular cages and two semiregular cages, obtained from the incidence graphs of an affine and a biaffine plane by generalizations of the reduction and graph amalgam operations from [M. Abreu, G. Araujo-Pardo, C. Balbuena, D. Labbate. Families of Small Regular Graphs of Girth 5. Discrete Math. 312(18) (2012) 2832–2842].

The vertex cover [formula omitted] problem in cubic graphs

A subset F of vertices of a graph G is called a vertex cover Pk set if every path of order k in G contains at least one vertex from F. Denote by ψk(G) the minimum cardinality of a vertex cover Pk set in G. The vertex cover Pk (VCPk) problem is to find a minimum vertex cover Pk set. In this paper, we restrict our attention to the VCP3 problem in cubic graphs. This paper proves that the VCP3 problem is NP-hard for cubic planar graphs of girth 3. Further we give sharp lower and upper bounds on ψ3(G) for cubic graphs and propose a 1.57-approximation algorithm for the VCP3 problem in cubic graphs.

Biregular Cages of Girth Five

Let $2 \le r < m$ and $g$ be positive integers. An $(\{r,m\};g)$-graph (or biregular graph) is a graph with degree set $\{r,m\}$ and girth $g$, and an $(\{r,m\};g)$-cage (or biregular cage) is an $(\{r,m\};g)$-graph of minimum order $n(\{r,m\};g)$. If $m=r+1$, an $(\{r,m\};g)$-cage is said to be a semiregular cage.In this paper we generalize the reduction and graph amalgam operations from [M. Abreu, G. Araujo-Pardo, C. Balbuena, D. Labbate. Families of Small Regular Graphs of Girth $5$. Discrete Math. 312 (2012) 2832--2842] on the incidence graphs of an affine and a biaffine plane obtaining two new infinite families of biregular cages and two new semiregular cages. The constructed new families are $(\{r,2r-3\};5)$-cages for all $r=q+1$ with $q$ a prime power, and $(\{r,2r-5\};5)$-cages for all $r=q+1$ with $q$ a prime. The new semiregular cages are constructed for $r=5$ and $6$ with $31$ and $43$ vertices respectively.

Gallai's Conjecture For Graphs of Girth at Least Four

AbstractIn 1966, Gallai conjectured that for any simple, connected graph G having n vertices, there is a path‐decomposition of G having at most paths. In this article, we show that for any simple graph G having girth , there is a path‐decomposition of G having at most paths, where is the number of vertices of odd degree in G and is the number of nonisolated vertices of even degree in G.

Computing the Girth of a Planar Graph in Linear Time

The girth of a graph is the minimum weight of all simple cycles of the graph. We study the problem of determining the girth of an n-node unweighted undirected planar graph. The first non-trivial algorithm for the problem, given by Djidjev, runs in O(n^{5/4} log n) time. Chalermsook, Fakcharoenphol, and Nanongkai reduced the running time to O(n log^2 n). Weimann and Yuster further reduced the running time to O(n log n). In this paper, we solve the problem in O(n) time.

A note on the upper bound and girth pair of [formula omitted]-cages

A (k;g)-cage is a k-regular graph of girth g with minimum order. In this work, for all k≥3 and g≥5 odd, we present an upper bound of the order of a (k;g+1)-cage in terms of the order of a (k;g)-cage, improving a previous result by Sauer of 1967. We also show that every (k;11)-cage with k≥6 contains a cycle of length 12, supporting a conjecture by Harary and Kovács of 1983.

On regular graphs of girth six arising from projective planes

In 1967, Brown constructed small k-regular graphs of girth six as induced subgraphs of the incidence graph of a projective plane of order q, q≥k. Examining the construction method, we prove that starting from PG(2,q), q=ph, p prime, there are no other constructions using this idea resulting in a (q+1−t)-regular graph of girth six than the known ones, if t is not too large (t≤p and roughly t<q1/6/8). Both algebraic and combinatorial tools are used.

Cayley Cages

A (k,g)-Cayley cage is a k-regular Cayley graph of girth g and smallest possible order. We present an explicit construction of (k,g)-Cayley graphs for all parameters k?2 and g?3 and generalize this construction to show that many well-known small k-regular graphs of girth g can be constructed in this way. We also establish connections between this construction and topological graph theory, and address the question of the order of (k,g)-Cayley cages.

Equitable colorings of planar graphs without short cycles

An equitable coloring of a graph is a proper vertex coloring such that the sizes of every two color classes differ by at most 1. Chen, Lih, and Wu conjectured that every connected graph G with maximum degree Δ≥2 has an equitable coloring with Δ colors, except when G is a complete graph or an odd cycle or Δ is odd and G=KΔ,Δ. Nakprasit proved that the conjecture holds for planar graphs with maximum degree at least 9. Zhu and Bu proved that the conjecture holds for every C3-free planar graph with maximum degree at least 8 and for every planar graph without C4 and C5 with maximum degree at least 7.In this paper, we prove that the conjecture holds for planar graphs in various settings, especially for every C3-free planar graph with maximum degree at least 6 and for every planar graph without C4 with maximum degree at least 7; this improves or generalizes results on equitable coloring by Zhu and Bu. Moreover, we prove that the conjecture holds for every planar graph of girth at least 6 with maximum degree at least 5.

Sparse graphs of girth at least five are packable

A graph is packable if it is a subgraph of its complement. The following statement was conjectured by Faudree, Rousseau, Schelp and Schuster in 1981: every non-star graph G with girth at least 5 is packable.The conjecture was proved by Faudree et al. with the additional condition that G has at most 65n−2 edges. In this paper, for each integer k≥3, we prove that every non-star graph with girth at least 5 and at most 2k−1kn−αk(n) edges is packable, where αk(n) is o(n) for every k. This implies that the conjecture is true for sufficiently large planar graphs.

Locally Symmetric Graphs of Girth 4

AbstractWe classify the family of connected, locally symmetric graphs of girth 4 (finite and infinite). They are all regular, with the exception of the complete bipartite graph . There are, up to isomorphism, exactly four such k‐regular graphs for every , one for , two for , and exactly three for every infinite cardinal k. In the last paragraph, we consider locally symmetric graphs of girth &gt;4.

On the Randić index and girth of graphs

The Randić index R(G) of a graph G is defined by R(G)=∑uv1d(u)d(v), where d(u) is the degree of a vertex u in G and the summation extends over all edges uv of G. In this work, we give a sharp upper bound and a lower bound of the Randić index among connected n-vertex graphs with girth g≥k(k≥3).