Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userBiregular cages of girth five
286
0
0.0
(
0
)
Ask ChatGPT about the research
No Arabic abstract
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 (2011) 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.
rate research
Read More
Let $q$ be a prime power; $(q+1,8)$-cages have been constructed as incidence graphs of a non-degenerate quadric surface in projective 4-space $P(4, q)$. The first contribution of this paper is a construction of these graphs in an alternative way by means of an explicit formula using graphical terminology. Furthermore by removing some specific perfect dominating sets from a $(q+1,8)$-cage we derive $k$-regular graphs of girth 8 for $k= q-1$ and $k=q$, having the smallest number of vertices known so far.
We show that the abelian girth of a graph is at least three times its girth. We prove an analogue of the Moore bound for the abelian girth of regular graphs, where the degree of the graph is fixed and the number of vertices is large. We conclude that one could try to improve the Moore bound for graphs of fixed degree and many vertices by trying to improve its analogue concerning the abelian girth.
We introduce the notion of a $[z, r; g]$-mixed cage. A $[z, r; g]$-mixed cage is a mixed graph $G$, $z$-regular by arcs, $r$-regular by edges, with girth $g$ and minimum order. In this paper we prove the existence of $[z, r ;g]$-mixed cages and exhibit families of mixed cages for some specific values. We also give lower and upper bounds for some choices of $z, r$ and $g$. In particular we present the first results on $[z,r;g]$- mixed cages for $z=1$ and any $rgeq 1$ and $ggeq 3$, and for any $zgeq 1$, $r=1$ and $g=4$.
Assume $ k $ is a positive integer, $ lambda={k_1,k_2,...,k_q} $ is a partition of $ k $ and $ G $ is a graph. A $lambda$-assignment of $ G $ is a $ k $-assignment $ L $ of $ G $ such that the colour set $ bigcup_{vin V(G)} L(v) $ can be partitioned into $ q $ subsets $ C_1cup C_2cupcdotscup C_q $ and for each vertex $ v $ of $ G $, $ |L(v)cap C_i|=k_i $. We say $ G $ is $lambda$-choosable if for each $lambda$-assignment $ L $ of $ G $, $ G $ is $ L $-colourable. In particular, if $ lambda={k} $, then $lambda$-choosable is the same as $ k $-choosable, if $ lambda={1, 1,...,1} $, then $lambda$-choosable is equivalent to $ k $-colourable. For the other partitions of $ k $ sandwiched between $ {k} $ and $ {1, 1,...,1} $ in terms of refinements, $lambda$-choosability reveals a complex hierarchy of colourability of graphs. Assume $lambda={k_1, ldots, k_q} $ is a partition of $ k $ and $lambda $ is a partition of $ kge k $. We write $ lambdale lambda $ if there is a partition $lambda={k_1, ldots, k_q}$ of $k$ with $k_i ge k_i$ for $i=1,2,ldots, q$ and $lambda$ is a refinement of $lambda$. It follows from the definition that if $ lambdale lambda $, then every $lambda$-choosable graph is $lambda$-choosable. It was proved in [X. Zhu, A refinement of choosability of graphs, J. Combin. Theory, Ser. B 141 (2020) 143 - 164] that the converse is also true. This paper strengthens this result and proves that for any $ lambda otle lambda $, for any integer $g$, there exists a graph of girth at least $g$ which is $lambda$-choosable but not $lambda$-choosable.
A emph{$[z, r; g]$-mixed cage} is a mixed graph $z$-regular by arcs, $r$-regular by edges, with girth $g$ and minimum order. %In this paper we study structural properties of mixed cages: Let $n[z,r;g]$ denote the order of a $[z,r;g]$-mixed cage. In this paper we prove that $n[z,r;g]$ is a monotonicity function, with respect of $g$, for $zin {1,2}$, and we use it to prove that the underlying graph of a $[z,r;g]$-mixed cage is 2-connected, for $zin {1,2}$. We also prove that $[z,r;g]$-mixed cages are strong connected. We present bounds of $n[z,r;g]$ and constructions of $[z,r;5]$-mixed graphs and show a $[10,3;5]$-mixed cage of order $50$.
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
