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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.
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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.
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.
We show that every n-vertex cubic graph with girth at least g have domination number at most 0.299871n+O(n/g)<3n/10+O(n/g).
An induced forest of a graph G is an acyclic induced subgraph of G. The present paper is devoted to the analysis of a simple randomised algorithm that grows an induced forest in a regular graph. The expected size of the forest it outputs provides a lower bound on the maximum number of vertices in an induced forest of G. When the girth is large and the degree is at least 4, our bound coincides with the best bound known to hold asymptotically almost surely for random regular graphs. This results in an alternative proof for the random case.
The first known families of cages arised from the incidence graphs of generalized polygons of order $q$, $q$ a prime power. In particular, $(q+1,6)$--cages have been obtained from the projective planes of order $q$. Morever, infinite families of small regular graphs of girth 5 have been constructed performing algebraic operations on $mathbb{F}_q$. In this paper, we introduce some combinatorial operations to construct new infinite families of small regular graphs of girth 7 from the $(q+1,8)$--cages arising from the generalized quadrangles of order $q$, $q$ a prime power.
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